Rough Isometries of Order Lattices and Groups

Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen

vorgelegt von Andreas Lochmann aus Bad Pyrmont

G¨ottingen 2009 Referenten der Dissertation: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D.

Mitglieder der Pr¨ufungskommission: Prof. Dr. Thomas Schick Prof. Laurent Bartholdi, Ph.D. Prof. Dr. Rainer Kress Prof. Dr. Ralf Meyer Prof. Dr. Ulrich Stuhler Prof. Dr. Andreas Thom

Tag der m¨undlichen Pr¨ufung: 6. August 2009 Gewidmet meinen Eltern

Preface

Assume you measure the distances between four points and find the following: One of the distances is 143, the distance between the other pair of points is 141, and all other distances are 99. You know that your error in measurement is not greater than 1. Is this result then compatible with the hypothesis, that the four points constitute a square in Euclidean space? Another example: You distribute five posts equidistantly along straight rails, then watch a train driving on the rails and take the times at which it passes the posts. The times you measure are: 0, 100, 202, 301, 401. However, the time the clocks show might deviate by up to 1 from the clock at the starting post. Are the measured times consistent with the assumption that the train has constant speed in Newtonian mechanics? These two puzzles are metaphors for a general problem in the application of mathematics. We have to use measurements that might contain errors to verify predictions of an ideal theory. If, for example, you find a function f between vector spaces to fulfill linearity only approximately, i.e. there is ǫ > 0 such that

d f(x + y), f(x) + f(y) ǫ ≤ for all x, y; can you ﬁnd a perfectly linear function g which is near f? Ulam stated this question during a talk around 1941, and Hyers found a positive solution shortly after by applying scaling arguments ([Hy]). Ulam and Hyers continued their collaboration, and were able to give positive answer in [HU] to the following question in 1947: Is every ǫ-isometric homomorphism of real- valued continuous function spaces in bounded distance to an isometry?

Theorem 1 (Hyers-Ulam) 1 Let K, K′ be compact metric spaces and let E and E′ be the spaces of all real valued continuous functions on K and K′, respectively. Let T (f) be a homomorphism of E onto E′ which is also an ǫ-isometry for some arbitrary, ﬁxed ǫ 0. Then there exists an isometric transformation U(f) of E onto E such ≥ ′ that U(f) T (f) 21ǫ for all f in E. In particular, the underlying metric || − || ≤ spaces K and K′ are homeomorphic.

This theorem evoked several similar results for linear functions ([Ra2]), but also about stability of diﬀerential equations, the stability of group actions ([GKM]), and approximate group homomorphisms, as deﬁned by Ulam in [U], section VI.1; see [HR] for a survey on this topic. 6

It is interesting to see that the isometry of the function spaces implies conditions of their underlying metric spaces—in this case they are homeomorphic. Our main interest during a large part of this thesis will be function spaces of Lipschitz functions, and their stability against rough isometries. Rough isometries are in some sense the prototype of approximative structure: the description of Hyers’ ǫ-linear functions would not be possible without a metric on the vector spaces. The idea simply is to replace an identity x = y by d(x, y) ǫ. In this sense, each description of approximations needs metric ≤ spaces. As the natural mappings of metric spaces are Lipschitz maps, isometries, and isometric embeddings, their rough counterparts are the most natural approximative tools, among them the rough isometry (or ǫ-isometry) as measure to describe the similarity of metric spaces. Our goal is to shed some light on the interplay of rough isometries with order lattice and group structures, but in particular to demonstrate their use in the study of the geometry of Lipschitz function spaces. There are detailed books and various articles on many aspects of Lipschitz function spaces, including isometries between them. A nice survey is Weaver’s book on Lipschitz Algebras [Wv] (cf. Section 2.6, where Weaver elaborates on the exact same questions we want to tackle here, but in a non-coarse context and under different conditions). However, to the knowledge of the author, no book nor article dealt with their coarse geometry yet. On the other hand, Lipschitz functions naturally appear in many aspects of coarse geometry, such as the Levy concentration phenomenon or the definition of Lipschitz-Hausdorff distance in [Gv2]. But they are not dealt with as metric spaces either. The structure of this thesis is as follows: We first give an overview of basic notions in handling with metric spaces and order lattices. We assume that the reader is familiar with metric spaces, so this part will just cover our use of infinities in metric spaces, the definition of hyperconvex and injective metric spaces, and those notions from coarse geometry which apply to our situation. The first chapter deals with a generalization of Birkhoff’s metric lattices to put the supremum metric on Lipschitz function spaces into a common context. During this chapter we derive several useful tools for to handle distances in Lipschitz function spaces, and introduce several metrics for Lipschitz functions, familiar ones as well as some more exotic distance functions. The final section in this chapter will define a very important notion for our analysis, a version of irreducibility which depends on the chosen metric of a lattice. We further see some first, general properties of this new kind of irreducibility. The second chapter will concentrate on Lipschitz function spaces with supremum metric, and we will examine the question, whether they are roughly isometric when their underlying spaces are. We first take a look at a simple, but efficient smoothening algorithm, then take a digression to demonstrate a corresponding Lemma for hyperconvex-space-valued Lipschitz functions. We then revive the situation of the first chapter, and define a rough version of isomorphism for intervaluation metric lattices, which fits well into the context of our new irreducibility-condition. By introducing minimal Lipschitz functions (which we call “Λ-functions”), we provide a lattice-theoretic base of the Lipschitz function space (see Definition 76), and put the smoothening re- Preface 7 sult of the first section into this more advanced context, re-proving it using new techniques:

Theorem 2 2 Let X,Y be metric spaces, ǫ 0. For each ǫ-isometry η : X Y , there is a + ≥ → 4ǫ-ml-isomorphism κ : Lip Y Lip X such that κ is ǫ-near f f η for all → 7→ ◦ f Lip Y (see Definitions 6, 29, 80, 89). ∈ We then continue to analyze our Λ-functions and find them to be exactly the irreducible elements of the first chapter. This allows us to state a converse of Theorem 2:

Theorem 3 3 Let X,Y be complete metric spaces and ǫ 0. For each ǫ-ml-isomorphism + ≥ κ : Lip Y Lip X there is an 88ǫ-isometry η : X Y , such that κ is 62ǫ-near → → f f η for all f Lip Y . 7→ ◦ ∈ As we make use of the order-lattice structure of the Lipschitz function spaces in our proofs, the following theorem by Kaplansky [Kp] is related to our results as well and we state it in its formulation by Birkhoﬀ ([Bi1], 2nd ed. p. 175f.):

Theorem 4 (Kaplansky) 4 Any compact Hausdorﬀ space is (up to homeomorphism) determined by the lattice of its continuous functions.

However, the proofs we present here not only make use of the lattice structure of Lip X, but of a metric on it as well. In this sense, the comparison with Kaplansky’s Theorem 4 is not perfect. We then ask whether it is possible to state Theorems 2 and 3 in a more general context, and give counter-examples to a broad range of different metrics on Lipschitz function spaces, among them the seemingly convenient L1-metric. We close this chapter with an example concerning quasi-isometries, and an application in the theory of scaling limits. The third chapter discusses the use of rough isometries in the context of finitely generated groups. One cannot overvalue the importance of coarse methods in Geometric Group Theory, which is in large parts based on the notion of quasi-isometry. We begin this chapter with another digression, demonstrating how some proofs can be reformulated in a rough context, while this is not possible with others. We then try to point out the possibilities rough isometries add to Geometric Group Theory, by providing a specifically rough isometry invariant (the exponential growth rate), and then show the connection between symmetries in some virtually abelian groups and rough isometries between them. We then continue with a theorem motivated by Bartholdi and translate it into the rough context in the final section, where we show that uniformly rough and quasi-isometries imply commensurability. Finally a remark to our notation: N0 denotes the non-negative integers, N∗ the positive integers, Cn denotes the cyclic group of order n. We sometimes drop function brackets where feasible. 8

Acknowledgements. The author wants to thank the “Graduiertenkolleg Gruppen und Geometrie” for supporting his research financially, through con- ferences and workshops, and by supporting the great open atmosphere at the institute. Best thanks go to Prof. Dr. Thomas Schick, Prof. Laurent Bartholdi, and Prof. Dr. Andreas Thom for their friendly support and great expertise, and to Prof. Dr. Themistocles Rassias for pointing the author at important works in connected fields. The author thanks his family and friends for always supporting him through the course of the doctorate, and in particular Dr. Johannes Härtel for being friend, critic, and guide.

Danksagung. Der Autor möchte dem “Graduiertenkolleg Gruppen und Geometrie” dafür danken, dass es seine Forschung sowohl finanziell als auch durch Konferenzen und Workshops und durch den Erhalt der großartigen At- mosphäre am Institut unterstützt hat. Bester Dank geht an Prof. Dr. Thomas Schick, Prof. Laurent Bartholdi, Ph.D., und Prof. Dr. Andreas Thom für ihre freundliche Unterstützung und große Expertise, sowie an Prof. Dr. Themistocles Rassias für die Verweise auf wichtige Arbeiten in verwandten Gebieten. Der Autor dankt seiner Familie und seinen Freunden für die stete Unterstützung im Laufe des Doktorats, und insbesondere Dr. Johannes Härtel für Freundschaft, Kritik und Anleitung.

Georg-August-Universit¨at G¨ottingen, Germany, 2009 eMail [email protected] Contents

0 Basic Notions 11 0.1 Basic Notions in Metric Spaces ...... 11 0.1.1 InﬁniteMetrics ...... 11 0.1.2 Injective and Hyperconvex Metric Spaces ...... 13 0.1.3 Basic Notions of Coarse Geometry ...... 16 0.2 Basic Notions in Order Lattices ...... 17

1 Order Lattices with Metrics 23 1.1 Valuation Metric Lattices ...... 23 1.2 Ultravaluation Metrics ...... 32 1.3 Intervaluation Metrics ...... 34 1.4 Complete Metric/Lattice-Irreducibility ...... 39

2 Rough Isometries of Lipschitz Function Spaces 43 2.1 SmootheningofLipschitzFunctions ...... 43 2.1.1 Hyperconvex and Non-Hyperconvex Codomains ...... 48 2.1.2 Algorithmic Aspects ...... 51 2.2 Roughml-Isomorphisms ...... 51 2.3 Λ-Functions...... 54 2.4 Λ-Functions are Completely ml-Irreducible ...... 59 2.5 InducingRoughIsometries...... 61 2.6 Other Metrics for Lip X ...... 65 2.7 Quasi-Isometries ...... 67 2.8 Scalinglimits ...... 68

3 Rough Isometries of Groups 71 3.1 TheTheoremofMazur-Ulam ...... 71 3.1.1 RoughAbelianness...... 75 3.2 CoarseRelationsforGroups...... 75 3.3 ExponentialGrowthRate ...... 77 3.4 TheAbelianCase...... 78 3.5 Rough Isometries of Quotients with Finite Kernel ...... 81 3.6 SharedIsometries...... 82 3.7 SharedRoughandQuasi-Isometries ...... 87

Bibliography 95 10 Chapter 0

Basic Notions

0.1 Basic Notions in Metric Spaces

0.1.1 Inﬁnite Metrics

As we will deal with lattice-theoretic notions, it is convenient to ensure the existence of a greatest possible distance in a metric space. To do so without having to restrict to bounded metric spaces, we will extend our notion of metric spaces to include inﬁnite distances.

Deﬁnition 5 ...... 5 We deﬁne the binary operation + on [0, ] to be as expected on [0, ), and ∞ ∞ set + z = z + = as well as z for all z [0, ]. We call ∞ ∞ ∞ ≤ ∞ ∈ ∞ z [0, ] positive if z = 0. ∈ ∞ 6

Deﬁnition 6 ...... 6 A pseudo-metric+ space (X, d) is a non-empty set X together with a mapping d : X X [0, ] which has the following properties: × → ∞

1. d(x, x) = 0 for all x X, and ∈ 2. d(x, z) d(x, y) + d(y, z) for all x, yz X. ≤ ∈

A pseudo-metric+ space (X, d) is a metric+ space if it is positive-deﬁnite, i.e.

3. d(x, y) = 0 implies x = y for all x, y X. ∈

The mapping d is then called a “metric”, “metric+ ”, or “distance”.

A (pseudo-)metric+ space (X, d) is a (pseudo-)ultrametric+ space, if it fulﬁlls

1. d(x, z) d(x, y) d(y, z) for all x, yz X, ≤ ∨ ∈ 12 Claim 8 where denotes the maximum. ∨ A (pseudo-)metric+ space (X, d) is called a true (pseudo-)metric space if d(x, y) = for all x, y X. This equals the common notion of a metric space in the 6 ∞ ∈ literature, like [BBI].

Deﬁnition 7 ...... 7 The diameter diam A of a non-empty subset A in a pseudo-metric+ space is the supremum of all distances of pairs of points in A.

A bounded (pseudo-)metric space is a (pseudo-)metric+ space X with ﬁnite diameter diam X < . ∞

Deﬁnition 8 ...... 8 An isometry [isometric embedding] is a bijection [injection] between pseudo-metric+ spaces which preserves the distance.

In most cases we call metrics just “ d ” without reference to the metric space, as it should be clear from the elements which metric is meant. We will make heavy use of the symbol

d(x,x′) d(y,y′) z − ≤

for some x, x′, y,y′ in X or Y and z [0, ]. To make sense of this in case one ∈ ∞ of the distances becomes inﬁnite, we deﬁne the former symbol to be equivalent to

d(x, x′) d(y, y′) + z and d(y, y′) d(x, x′) + z. ≤ ≤ In particular, we find = 0. This might seem unfamiliar. Note however, |∞ − ∞| that z z can be perfectly understood as a metric on [0, ] itself. | − ′| + ∞ Remark There is no non-trivial convergence to in this metric, is just an ∞ ∞ infinitely far away point. We will not need a convergence to infinity anyway; the addition of serves to obtain completeness of order lattices, particularly ∞ of [0, ] itself. ∞ Furthermore, it is obvious that a metric+ space X always is a disjoint union of true metric spaces X with d(x, x ) = if and only if x X , x X with j ′ ∞ ∈ j ′ ∈ k j = k, for j, k J. We call the X components of X. We call X complete, 6 ∈ j if all of its components are complete as true metric spaces, i.e. each Cauchy sequence converges.

The diﬀerence between metric+ and true metric spaces is rather small, as a metric+ space merely is a collection of true metric spaces. For example, the topology of a metric+ space is the disjoint union of the topology of its components, and any converging sequence eventually is contained in one of these components, with only ﬁnitely many exceptions at the beginning of the sequence. Basic Notions in Metric Spaces 13

Example 9 ...... 9 The algebra of continuous functions f : R R together with the supremum → metric

d (x, y) := sup f(x) f(y) ∞ x R − ∈ is a metric+ space. Its components are those functions with ﬁnite distance to each other, so the subset of bounded functions constitutes the component of the zero function, and the identity belongs to another component.

Example 10 ...... 10 Similar to the usual definition of a norm on a vector space, it is possible to define a norm+ with possibly infinite value, and derive a metric+ from it. For example, the true metric space of square-summable complex sequences is a component of the metric+ space of all complex sequences with norm+

(x , x ,...) := x 2 [0, ] 1 2 2 | j| ∈ ∞ sj N∗ X∈ for any complex sequence (x1, x2,...).

Definition 11 ...... 11 The Hausdorff-distance of two subsets A and B in a pseudo-metric+ space is the infimum of all non-negative numbers r such that for each x A there is ∈ y B with d(x, y) r and vice versa. ∈ ≤ A subset A of a pseudo-metric+ space is ǫ-dense in A, if its Hausdorff-distance to A is less than or equals ǫ.A dense subset is a 0-dense subset. A roughly dense subset is a subset for which its Hausdorff-distance to A is finite.

Deﬁnition 12 ...... 12 A pseudo-metric+ space X is separable if it contains a countable dense subset.

0.1.2 Injective and Hyperconvex Metric Spaces

Deﬁnition 13 ...... 13 A short map (sometimes called a “metric map”) between pseudo-metric+ spaces X and Y is a 1-Lipschitz map π : X Y , this means it fulﬁlls d (πx, πy) → Y ≤ d (x, y) for all x, y X. X ∈ Short maps are the canonical morphisms to construct a category Met of true metric spaces, as a bijective short map whose inverse is short as well is an isometry. They form morphisms in the category Met+ of metric+ spaces as well, as short maps map componentwise.

Deﬁnition 14 ...... 14 An injective metric+ space X is an injective object in the category Met+, i.e. 14 Claim 18 for each metric space Y and short map f : Y X, and each short embedding + → i : Y ֒ Z of Y into another metric space Z, there is a short map g : Z X → + → which extends f (f = g i). ◦ Proposition 15 ...... 15 A metric+ space is injective if and only if all of its components are injective. In particular, a true metric space is injective in Met if and only if it is injective in Met+.

Proof Note that a short map maps a component into a single component of the codomain, i.e. the map which assigns each metric+ space its set of components is a functor. “ ” Let X be a component of the injective metric space X. Let f : Y X ⇒ 1 + → 1 be a short map, and Y Z metric spaces. Y must be a true metric space. ⊆ + Let Z be the component of Y in Z. Extend f to the whole of X, its image 1 |Z1 must still be in the same component X . Now choose an arbitrary point x X 1 0 ∈ 1 and map all remaining components of Z to this point. This yields an extension g : Z X of f. → 1 “ ” Let Y Z, f : Y X, where X is componentwise injective. Choose a ⇐ ⊆ → point x X. Extend f componentwise on each component of Z which inter- 0 ∈ sects Y ; map each remaining component of Z to x0.

As next, we state the classical result that metric spaces are injective if and only if they are hyperconvex. This accounts for true metric spaces as well as for metric+ spaces. Hyperconvexity is a stronger form of convexity.

Deﬁnition 16 ...... 16 Let X be a metric+ space. X is convex in the sense of Menger ([Me], p. 81) if for each x,y X there is z X r x, y such that d(x, y)= d(x, z)+ d(z, y). ∈ ∈ { } X is totally convex (or convex in the sense of [EK]) if for all a, b (0, ] with ∈ ∞ d(x, y) a + b there is z X with d(x, z) a and d(z, y) b. ≤ ∈ ≤ ≤ X is hyperconvex if for any family of points (xj)j J X and numbers rj [0, ] ∈ ⊆ ∈ ∞ with d(x , x ) r + r for all j J, with J an arbitrary index set, there is j k ≤ j k ∈ an element z X with d(x , z) r for all j J. In simpler words: If any ∈ j ≤ j ∈ family of balls could theoretically intersect pairwise (given their radii), they all intersect.

Example 17 ...... 17 Typical examples for hyperconvex true metric spaces are (real) trees and Rn with L∞-metric as well as certain subsets thereof. In particular, R and each real interval are hyperconvex.

Proposition 18 ...... 18 A metric+ space X is hyperconvex if and only if all of its components are hyperconvex. Basic Notions in Metric Spaces 15

Proof All xj with infinite rj can be neglected. On the other hand, if two points x ,x X are in different components, at least of one r or r must be j k ∈ j k infinite, hence the hyperconvexity property always reduces to hold on a single component of X. On the other hand, assume X is hyperconvex. Given a family of balls with center x and radius r in one component X X, we may use j j 1 ⊆ hyperconvexity in X to gain the desired element z X. As long as at least two ∈ of the r were finite, we must choose z X . If only one r is finite, we may j ∈ 1 j choose z = xj.

To provide a feeling for hyperconvex and injective metric+ spaces, we cite some assorted theorems without giving proofs. Due to Propositions 15 and 18, there are no obstructions to generalize the original statements to metric+ spaces.

Theorem 19 ([AP], Theorem 2.4) 19 A metric+ space is hyperconvex if and only if it is injective.

Theorem 20 ([EK], Theorem 3.1) 20 A product of arbitrarily many hyperconvex metric+ spaces with supremum metric+ is hyperconvex. (This statement is much easier in the Met+ form than the original statement for true metric spaces.)

Theorem 21 ([EK], Proposition 3.2) 21 Any hyperconvex metric+ space is complete.

Theorem 22 (Baillon’s Theorem; [EK], Theorem 5.1) 22 The intersection of a descending chain of non-empty hyperconvex subsets of a bounded metric space is non-empty and hyperconvex.

Theorem 23 ([EK], Theorem 6.1) 23 The ﬁxed point set of a short map f : X X acting on a hyperconvex bounded → metric space X is non-empty and hyperconvex.

Theorem 24 ([EK], Theorem 9.6) 24 Let X be hyperconvex, K > 0. The family of all bounded K-Lipschitz functions X X with supremum metric is hyperconvex. → Theorem 25 ([I], Theorem 2.1) 25 Each true metric space X has an injective envelope (i.e. a minimal injective space into which X isometrically embeds), which is unique up to isometry.

Theorem 26 ([I], Remark 2.11) 26 The injective envelope of a compact space is compact.

Theorem 27 ([AP], Theorem 3.3) 27 If X is a metric space, A X hyperconvex, B X such that A and B have + ⊆ ⊆ identical closures in X, then B is hyperconvex as well. 16 Claim 31

Theorem 28 ([AP], Theorem 3.9) 28 Let X be hyperconvex. Then A X is hyperconvex if and only if it is a retract ⊆ of X by a contracting retraction.

0.1.3 Basic Notions of Coarse Geometry

A well written introduction to coarse geometry is the book by Burago, Burago and Ivanov ([BBI]). In the following, let X and Y be metric+ spaces.

Deﬁnition 29 ...... 29 Two (set theoretic) mappings α, β : X Y are ǫ-near to each other, ǫ 0, if → ≥ d (αx,βx) ǫ x X. (We drop brackets where feasible.) Y ≤ ∀ ∈ A (set theoretic) mapping α : X Y is ǫ-surjective, ǫ 0, if for each y Y → ≥ ∈ there is x X such that d (αx,y) ǫ. ∈ Y ≤

Deﬁnition 30 ...... 30 A (not neccessarily continuous) map η : X Y is called a (λ, ǫ)-quasi- → isometric embedding, ǫ, λ 0 (which shall always imply λ, ǫ R), if ≥ ∈ 1 λ− d (x, x′) ǫ d (ηx, ηx′) λ d (x, x′) + ǫ X − ≤ Y ≤ X for all x,x X. ′ ∈ A pair η : X Y , η : Y X of (λ, ǫ)-quasi-isometric embeddings is called a → ′ → (λ, ǫ)-quasi-isometry if η η and η η are ǫ-near the identities on Y and X, ◦ ′ ′ ◦ respectively. When we speak of a “quasi-isometry η : X Y ” a corresponding → map η′ shall always be implied. X and Y are called quasi-isometric, if there is a quasi-isometry between them.

Deﬁnition 31 ...... 31 A (1, ǫ)-quasi-isometric embedding, ǫ 0, is called a ǫ-isometric embedding. It ≥ fulﬁlls

d (x,x′) d (ηx,ηx′) ǫ | X − Y | ≤ for all x,x X. ′ ∈ An ǫ-isometry is a (1, ǫ)-quasi-isometry. The map η : X Y is called a rough → isometry if there is some ǫ 0 such that η is an ǫ-isometry. ≥ X and Y are called ǫ-isometric [roughly isometric], if there is an ǫ-isometry [any ǫ-isometry] between them.

Historical Remark It is diﬃcult to attribute the concept of rough isometry to a single person, as it was always present in the notion of quasi-isometry, which itself was an obvious generalization of what was then called pseudo-isometry by Mostow in his 1973-paper about rigidity (see [Mo], [Gv1], [Kn]). Recent developments about the stability of rough isometries can be found in [Ra1]. Basic Notions in Order Lattices 17

Deﬁnition 32 ...... 32 A mapping f : X Y is called (K, ǫ)-Lipschitz (i.e. “K-Lipschitz map on → ǫ-scale” in [Gv3]), ǫ, K 0, iﬀ ≥ d (f(x), f(y)) K d (x, y) + ǫ x, y X. Y ≤ · X ∀ ∈

If ǫ = 0, f is K-Lipschitz (continuous). Deﬁne LipK,ǫ(X, Y ) to be the set of all (K, ǫ)-Lipschitz functions X Y , and Lip X := Lip (X, [0, ]), → K,ǫ K,ǫ ∞ Lip X := Lip1,0(X).

If nothing else is said, [0, ] = R 0 is the default codomain for a ∞ ≥ ∪ {∞} Lipschitz function. Assume f to be a (K, ǫ)-Lipschitz function on X and f(x) = for some ∞ x X. Then clearly f(y) = for all y in ﬁnite distance to x. Thus, if X is a ∈ ∞ true metric space, we have Lip X = Lip(X, R 0) . ≥ ∪ {∞}

0.2 Basic Notions in Order Lattices

Good starting points for Lattice Theory are Gr¨atzer ([Gr]) and the classical book by Birkhoﬀ ([Bi1]).

Deﬁnition 33 ...... 33 A lattice (L, , ) is a set L together with two mappings , : L L L ∧ ∨ ∧ ∨ × → which are commutative, associative and fulﬁll the absorption laws f (f g)= ∧ ∨ f (f g)= f for all f, g L. A lattice is a partially ordered set by ∨ ∧ ∈

by absorption f g : f g = f f g = g ≤ ⇔ ∧ ⇔ ∨

L is distributive if and distribute over each other. ∨ ∧ L is bounded (as a lattice) if there exists a smallest element 0 L and a largest ∈ element 1 L. ∈ L is complete if all inﬁma and all suprema of all subsets of L exist in L. (A complete lattice always is bounded.) L is complemented if L is bounded and for each f L there is a complement ∈ g L such that f g = 1 and f g = 0. ∈ ∨ ∧ L is a Boolean lattice if it is distributive and complemented.

Example 34 ...... 34 Let X be some set, and let L be any family of subsets of X closed under and ∩ . Then (L, , ) is a distributive lattice, sometimes called a “ring of sets” (we ∩ ∩ ∪ will stick to “lattice of sets”). If for all A L the complement X r A L as ∈ ∈ well, then (L, , ) is a Boolean lattice (“ﬁeld of sets”). ∩ ∪ 18 Claim 38

Example 35 ...... 35 Let X be a topological space, and T the family of all open sets in X. T is a lattice by union and intersection, bounded by and X, distributive (as it is a ∅ lattice of sets). However, if X is Hausdorﬀ, then T is complete if and only if T is the discrete topology, and if and only if T is complemented.

Deﬁnition 36 ...... 36 Let (L, , ) be a lattice. ∧ ∨ An element p L is join-irreducible if, whenever p = f g with f, g L, then ∈ ∨ ∈ p = f or p = g. An element p L is join-prime if, whenever p f g with f, g L, then p f ∈ ≤ ∨ ∈ ≤ or p g. ≤ An element p L is completely join-irreducible if, whenever p = f , with ∈ j J j f L, then p = f for some j J, J an arbitrary index set. Same∈ for com- j ∈ j ∈ W pletely join-prime. A lower set in a partially ordered set P is a subset Q of P with: f g, g Q, ≤ ∈ f P implies f Q. ∈ ∈ A sublattice of L is a subset of L closed under and . ∧ ∨ An ideal is a sublattice I L such that f g I whenever f L and g I ⊆ ∧ ∈ ∈ ∈ (equivalent deﬁnition: a sublattice which is a lower set). A proper ideal is an ideal I which is a proper subset of L. A principal ideal is an ideal I which is generated by a single element. A prime ideal is a proper ideal P of a Boolean algebra for which holds: If f, g L ∈ and f g P , then f P or g P . The family of all prime ideals in L is called ∧ ∈ ∈ ∈ P (L).

Proposition 37 ...... 37 In a distributive lattice, join-irreducibility and join-primeness are equivalent ([Bi1], Lemma III.3.1).

Proof Let p L be join-prime. Then it is join-irreducible by deﬁnition. Now ∈ let p L be join-irreducible and p f g with f, g L. We ﬁnd ∈ ≤ ∨ ∈ p = p (f g) = (p f) (p g). ∧ ∨ ∧ ∨ ∧ As p is join-irreducible, we have p = p f, this is p f, or p = p g, which ∧ ≤ ∧ means p g. ≤

Example 38 ...... 38 We note that a prime ideal is not the same as a principal ideal generated by a join-prime element: The positive integers equipped with least common divisor and greatest common multiple constitute a distributive and unbounded lattice L = (N∗, gcd, lcm) Basic Notions in Order Lattices 19 with 1 as least element. Join-irreducible/join-prime elements are exactly the prime powers. The subset A = 5, 15, 50, 150 is an example for a sublattice { } in L. The least ideal encompassing A is the subset of all divisors of 150. It is a principal ideal generated by 150. The principal ideal which is generated by a join-irreducible element pn, p prime, is just 1, p, p2,..., pn . The subset { } B of all powers of 7 is a non-principal proper ideal. It is not a prime ideal: gcd(14, 35) = 7, but neither 14 B nor 35 B. The subset C = Zr 7Z of ∈ ∈ all positive integers except the multiples of 7 is a prime ideal: If 7 ∤ gcd(f, g), then 7 ∤ f or 7 ∤ g. On the other hand, the subset Zr 6Z is not even an ideal.

Example 39 ...... 39 Each ideal in a lattice is a lower set, but not vice versa: In the lattice L = (N , gcd, lcm) the subset Q := 1, 5, 25, 29 is a lower set, but not an ideal, ∗ { } because lcm(5, 29) = 145 / Q. ∈

Example 40 ...... 40 Let X be any set of at least two elements, and consider its powerset (L = ℘(X), , ). Its join-irreducibles are the one-element subsets and the empty ∩ ∪ set. A principal ideal generated by A X is ℘(A) ℘(X). Given any x X ⊆ ⊆ ∈ consider := A X : x / A . is a prime ideal. Fx { ⊆ ∈ } Fx

In Lipschitz function spaces, principal ideals which are generated by a join- irreducible are more interesting, as join-irreducibles tend to interact in more interesting ways: Their intersections can be non-trivial.

Theorem 41 41 (Representation Theorem for Distributive Lattices, G. Birkhoﬀ 1933, M. H. Stone 1936; [Gr], Th. II.1.19) A lattice is distributive if and only if it is isomorphic to a lattice of sets.

Proof Birkhoff proved a very similar representation theorem for finite distributive lattices in [Bi1] (see next Theorem), Stone later extended the proof to infinite distributive lattices as well. In addition, he showed that a complemented distributive lattice is isomorphic to a complemented lattice of sets. The main idea of the proof is to show that the map

π : L ℘(P (L)) → f p P (L) : f / p 7→ { ∈ ∈ } between L and the power set of the set P (L) of all prime ideals in L is an injective homomorphism, and hence π constitutes an isomorphism between |im π L and a sublattice of ℘(P (L)).

Subsequently, each distributive lattice is isomorphic to a sublattice of a powerset. Today there is a broad variety of representation theorems for (distributive) lattices, particularly as lattices of open, closed or clopen subsets in topological 20 Claim 45 spaces. These theorems are subsumed under the term “Stone-type dualities”. An earlier version is the following theorem by Birkhoﬀ for ﬁnite distributive lattices. We will not make use of it, but its main idea of reconstructing a distributive lattice solely from its join-irreducibles will be a major theme in Chapter 2.

Theorem 42 (Birkhoﬀ’s Representation Theorem) 42 Let L be any ﬁnite distributive lattice, and J L the subset of join-irreducibles. ⊆ J is a partially ordered set (not a sublattice!). The family of lower sets in J is a lattice of sets, and isomorphic to L ([Bi1], Corollary III.3.2).

Of particular interest for us is L = Lip X for some metric space X, with and + ∧ pointwise minimum and maximum respectively, and , pointwise inﬁmum ∨ and supremum. Lip X is a distributive lattice, as the distributivity is inherited V W from (R, , ). The following proposition is a special case of Lemma 6.3 in ∧ ∨ [He] and Proposition 1.5.5 in [Wv].

Proposition 43 ...... 43 Let X be a metric+ space. Then Lip X is complete as a lattice.

Proof Let J be some arbitrary index set, and let f be in Lip X for each j J. j ∈ Obviously, [0, ] is complete as a lattice, with = and = 0. So we ∞ ∞ deﬁne pointwise ∅ ∅ V W

g(x) := fj(x), h(x) := fj(x) j J j J _∈ ^∈ and observe that g and h : X Z are Lipschitz: For arbitrary x,y X holds → ∈ h(x) f (x) f (y) + d(x,y) ≤ j ≤ j for all j J, and thus, by passing to the inﬁmum: ∈ h(x) h(y) + d(x,y). ≤ Same for g.

Example 44 ...... 44 The space C([0, 1], [0, 1]) of real-valued continuous functions [0, 1] [0, 1] is → not a complete lattice with pointwise minimum and maximum: Choose fj(x)= 0 (1 j x), j N , the inﬁmum is not continuous. The same example shows ∨ − · ∈ ∗ that the space K 0 LipK,0 X of all Lipschitz-functions with arbitrary Lipschitz constant is not a complete≥ lattice. S

Example 45 ...... 45 Besides its Stone representation, we want to provide another, more intuitive Basic Notions in Order Lattices 21 representation of Lip X by a lattice of sets, using its hypograph (cp. “epigraph” in [Ro])

hyp : Lip X ℘ (X [0, ]) → × ∞ f (x, r) : f(x) r . 7→ { ≤ } (im hyp, , ) obviously is isomorphic to (Lip X, , ) as a lattice; however, ∩ ∪ ∧ ∨ they are not yet isomorphic as complete lattices: Inﬁnite unions of the closed sets in im hyp are not closed in general – we have to use the union with closure “ ¯” instead of the traditional union. (Alternatively, we could identify subsets ∪ of X [0, ] with the same closure.) × ∞ 22 Claim 45 Chapter 1

Order Lattices with Metrics

1.1 Valuation Metric Lattices

For the most part of this thesis, we will consider the supremum metric+

d (f, g) := f(x) g(x) . ∞ − x X _∈ on Lip X. But before we come to further investigate this, we present Birkhoff’s [Bi1] definition of a metric lattice and demonstrate the difference to our situation.

Deﬁnition 46 ...... 46 A valuation on a lattice L is a function v : L R which satisﬁes the modular → law

v(f) + v(g) = v(f g) + v(f g) f, g L. ∧ ∨ ∀ ∈ A valuation v on L is called isotone [positive] if for all f, g L the relation ∈ f < g implies v(f) v(g) [v(f) < v(g)]. ≤ If L is totally ordered, then each function v : L R is a valuation. It is isotone → [positive] if and only if v is [strictly] monotonically increasing.

Example 47 ...... 47 Let L = (N∗, gcd, lcm). Then each logarithm is a positive valuation on L.

Example 48 ...... 48 Let V be any ﬁnite dimensional vector space, and L = PG(V ) its lattice of sub- vector spaces, with the intersection and the span (the projective geometry ∧ ∨ of V ). Then the dimension function is a positive valuation on L.

Historical Remark The theory of valuations has been mainly developed and popularized by John von Neumann and Garrett Birkhoff. In the early years of 24 Claim 49 the 1930s, von Neumann worked on a variation of the ergodic hypothesis, and inadvertently competed with George David Birkhoff. Only some years later, his son Garrett Birkhoff pointed von Neumann at the use of lattice theory in Hilbert spaces. He wrote about this in a note of the Bulletin of the AMS in 1958 [Bi2].

John von Neumann’s brilliant mind blazed over lattice theory like a meteor, during a brief period centering around 1935–1937. With the aim of interesting him in lattices, I had called his attention, in 1933– 1934, to the fact that the sublattice generated by three subspaces of Hilbert space (or any other vector space) contained 28 subspaces in general, to the analogy between dimension and measure, and to the characterization of projective geometries as irreducible, finite- dimensional, complemented modular lattices. As soon as the relevance of lattices to linear manifolds in Hilbert space was pointed out, he began to consider how he could use lattices to classify the factors of operator-algebras. One can get some impression of the initial impact of lattice concepts on his thinking about this classification problem by reading the introduction of [...], in which a systematic lattice-theoretic classification of the different possibilities was initiated. [...] However, von Neumann was not content with considering lattice theory from the point of view of such applications alone. With his keen sense for axiomatics, he quickly also made a series of funda- mental contributions to pure lattice theory.

The modular law in its earliest form (as dimension function) appears in two papers from 1936 by Glivenko and von Neumann ([Gl], [vN]). Von Neumann used it (and lattice theory in general) in his paper to deﬁne and study Continu- ous Geometry (aka. “pointless geometry”), and later applied his knowledge to found Quantum Logic in his Mathematical Foundations of Quantum Mechanics. A later survey about metric posets is [Mn].

Example 49 ...... 49 Let (X, Σ,µ) be a probability space. The σ-algebra Σ is a Boolean lattice by union and intersection. Let c R be arbitrary, then ∈ v(A) := µ(A)+ c

deﬁnes an isotone valuation on Σ with v( ) = c. The valuation v is positive if ∅ and only if there are no null sets in X other than . ∅ Proof: Let A = be a null set. Then ( A, but µ( )=0= µ(A). Conversely, 6 ∅ ∅ ∅ if A ( B Σ, and v(A)= v(B), then deﬁne C := B r A. B is a disjoint union ∈ of A and C, so by σ-additivity of µ holds µ(A) = µ(B) = µ(A)+ µ(C), hence µ(C) = 0. Valuation Metric Lattices 25

The distance function d (A, B) := v(A B) v(A B) (whose properties will v ∪ − ∩ be proved in Lemma 53) is the measure of the symmetric diﬀerence A △ B of A and B, if A △ B Σ. It relates to the Hausdorﬀ distance just as the 1-distance ∈ of functions relates to the supremum distance.

We further exploit the connection between dv and the symmetric diﬀerence. The symmetric diﬀerence as an operation makes sense only in complemented lattices (power sets are examples for complemented lattices). But although all distributive lattices can be represented by a lattice of sets, and hence can be embedded into a complemented lattice, they need not be complemented by themself.

Example 50 ...... 50 Let L = (Z, min, max). A possible representation of L is the family

A(L) := ( ,n] Z n Z { −∞ ∩ | ∈ } of subsets of Z, equipped with union and intersection. This lattice is order- isomorphic to L. The canonical distance in Z is given by the L-valuation v(n) := n, as well as by the counting measure of the symmetric difference of sets in A(L). However, L is distributive, but not complemented, there is no probability measure µ on A(L) which corresponds to v, and the symmetric difference of two sets in A(L) does not yield a set of A(L) again; these problems are all connected to each other, as they all base on the fact that L is not complemented. They still persist even if one completes L to L := (Z , min, max) and ′ ∪ {±∞} generalizes v to a valuation with infinity.

Note that the existence of symmetric diﬀerences f △ g, of complements f c, and of diﬀerence sets f r g are equivalent to each other in complete lattices of sets:

f c := 1 △ f = 1 r f with 1 := f f L _∈ f △ g := (f g)c (f g) = (f r g) (g r f) ∧ ∧ ∨ ∪ f r g := f gc = f △ (f g) ∧ ∧ In particular, complement, symmetric difference, and difference are unique. Lip X never is a complemented lattice. Hence, there is no difference defined on Lip X; however, given a valuation v and f, g Lip X, we may define a difference ∈ valuation of f and g by:

w(f, g) := v(f) v(f g). − ∧ As f g f, we conclude that for positive or isotone v, the diﬀerence valuation ∧ ≤ w always is non-negative.

Proposition 51 (Diﬀerence valuation properties) ...... 51 Let v be an isotone valuation on a distributive lattice L and w its diﬀerence valuation. Then the following holds for all f,g,h L: ∈ 26 Claim 51

1. w(f, g) + w(g, f) = v(f g) v(f g) =: d (f, g) ∨ − ∧ v 2. w(f, g) = w(f, g h) + w(f h, g) (cut law) ∨ ∧ 3. f g w(f, g) = 0 ≤ ⇒ 4. v is positive if and only if for all f, g L holds: w(f, g) = 0 f g. ∈ ⇔ ≤ 5. Let w : L L R be a map satisfying property (2), c R arbitrary, and × → ∈ let 0 L be a least element. Then v(f) := w(f, 0) + c is a valuation with ∈ w as difference valuation, and all valuations with w as difference valuation are of this form. If w(f, g) 0 for all f, g L, then v is isotone. ≥ ∈ Proof (1) Simply by the defining property of v: w(f, g)+ w(g, f) = v(f) 2v(f g)+ v(g) − ∧ = v(f g) v(f g) ∨ − ∧ (2) By the defining property of v we know: v(f h)+ v(f g) = v ((f g) (f h)) + v ((f g) (f h)) ∧ ∧ ∧ ∨ ∧ ∧ ∧ ∧ = v (f (g h)) + v(f g h) ∧ ∨ ∧ ∧ Inserting this into the definition of w yields: w(f, g) = v(f) v(f g) − ∧ = v(f) v (f (g h)) + v(f h) v(f g h) − ∧ ∨ ∧ − ∧ ∧ = w(f, g h)+ w(f h, g) ∨ ∧ We call this equation “cut law” in view of its meaning in Venn diagrams (see Figure 1.1). (3) Using (2): f = f g w(f, g)= w(f g, g)+ w(f, g g) = 2 w(f, g). ∧ ⇒ ∧ ∨ (4, “ ”) f g is equivalent to f g = f, and by positivity, equivalent to ⇒ ≤ ∧ v(f g)= v(f). ∧ (4, “ ”) Let f < g. Then f = g f, and ⇐ ∧ v(g) v(f) = v(g) v(g f) = w(g,f). − − ∧ By isotony we know v(f) v(g), assume v(f) = v(g), then w(g,f) = 0, and ≤ hence g f, contradicting f < g. ≤ (5) First of all we deduce an easy consequence of properties (2) and (3): w(f g, g) = w((f g) f, g)+ w(f g,f g) = w(f, g) ∨ ∨ ∧ ∨ ∨ With this at hand, we can easily conclude that v is a valuation and that w is its difference valuation: v(f) = w(f, 0) + c = w(f g, 0) + w(f, g)+ c ∧ = v(f g) + w(f, g) ∧ and v(f g) = w(f g, 0) + c = w(g, 0) + w(f g, g) + c ∨ ∨ ∨ = v(g) + w(f, g) v(f) + v(g) = v(f g) + v(f g) ⇒ ∨ ∧ Valuation Metric Lattices 27

f g f g f g

h h h w(f, g)= w(f, g v h) + w(f v h, g)

Figure 1.1: Visualization of the cut law of diﬀerence valuations (Proposition 51.2), using Venn diagrams. Using a representation π, the set π(f) r π(g) is cut along π(h) to give π(f) r π(g h) and π(f h) r π(g). ∨ ∧

Let w(f, g) 0, f g, then v(g) v(f)= v(g) v(f g) 0, i.e. v is isotone. ≥ ≤ − − ∧ ≥ Now let v be any valuation with w as diﬀerence valuation, then

v(f) = v(f) v(f 0) + v(0) = w(f, 0) + v(0) − ∧ obviously holds, choose c = v(0).

Proposition 51.5 shows the equivalence of the concepts of valuation and difference valuation for complete lattices, so we may deﬁne the term “diﬀerence valuation” without reference to an actual valuation:

Definition 52 ...... 52 A difference valuation on a distributive lattice L is a function w : L L R × → which satisfies the cut law

w(f, g) = w(f, g h) + w(f h, g). ∨ ∧ A diﬀerence valuation w is called isotone if its values are non-negative, and positive, if w(f, g) = 0 implies f g. ≤

The following Lemma is a part of Theorem X.1 and a note in subsection X.2 of [Bi1], and can equally well be stated in terms of valuations as well as diﬀerence valuations:

Lemma 53 ...... 53 Let v be an isotone valuation on the distributive lattice L. Then

d (f, g) := v(f g) v(f g) v ∨ − ∧ deﬁnes a pseudo-metric with the following properties:

1. If there is a least element 0 L, then ∈ v(f) = v(0) + d (f, 0) for all f L, v ∈ 28 Claim 53

f g f g f g f g f g

d(f, g) = + + + h h h h h

f g f g f g

d(f, h) = + h h h

f g f g f g

d(h, g) = + h h h

Figure 1.2: Proof of the triangle inequality for valuation metric lattices, e.g. of Lipschitz function spaces with L1-metric. Note that f,g,h are functions in this case, and symbolized by sets via Stone duality.

2. dv is a metric if and only if v is positive.

We call dv a valuation (pseudo-)metric. A lattice together with a valuation metric is sometimes called a metric lattice; however, as we will deal with lattices with non-valuation metrics as well (particularly the supremum metric), we should better distinguish between valuation metric lattices and non-valuation metric lattices.

Proof dv(f,f) = 0, dv(f, g) = dv(g,f) and property (1) are obvious. The absorption laws tell us that f f g and f f g for all f, g L, hence ≤ ∨ ≥ ∧ ∈ f g f g and isotony of v yield d (f, g) 0. Contrary to Birkhoﬀ, we will ∨ ≥ ∧ v ≥ use diﬀerence valuations to prove triangle inequality:

d (f, g) = w(f, g h)+ w(f h, g)+ w(g, f h)+ w(g h, f) v ∨ ∧ ∨ ∧ Due to positivity of w and property (2) in Proposition 51, we have

w(f, g h) w(f, h) ∨ ≤ w(f h, g) w(h, g) ∧ ≤ w(g,f h) w(g, h) ∨ ≤ w(g h, f) w(h, f) ∧ ≤ And thus

d (f, g) w(f, h)+ w(h, f)+ w(h, g)+ w(g, h) d (f, h)+ d (h, g). v ≤ ≤ v v

We ﬁnally show that dv is a metric if and only if v is positive. Again, we use dv(f, g) = w(f, g)+ w(g,f) to see that dv(f, g) = 0 implies w(f, g) = 0 and w(g,f) = 0. Proposition 51, property (4) applies: f g and g f, thus ≤ ≤ Valuation Metric Lattices 29

f g f g f g

h h h

v d(f, g)= d(f v h, g v h) + d(f v h, g h)

f g f g f g

h h h

d(f, g)= d(f v (g v h), f (g v h))

v + d(g v (f v h), g (f v h))

f g f g f g

v d(f, g)= d(f, f g) + d(f v g, g)

=d(g, f v g) + d(f v g, f)

Figure 1.3: Demonstration of the use of Venn diagrams to prove calculation rules in valuation metric lattices. The final example shows how a single Venn diagram can be interpreted in two different ways to match two different rules. f = f g = g. For the other direction, keep in mind that f g w(f, g) = 0 ∧ ≤ ⇒ always holds. So, assume that dv is a metric, and w(f, g) = 0 though f g. Then f = f g, d (f,f g) > 0, and w(f g,f)= d (f,f g) w(f, g) > 0. 6 ∧ v ∧ ∧ v ∧ − But f g f, so w(f, g) = 0, contradiction. ∧ ≤

The Lemma still holds for non-distributive lattices, just property (1) is weaker: d (f g,f h) + d (f g,f h) d(g, h) for all f,g,h L. v ∨ ∨ v ∧ ∧ ≤ ∈ The proof uses the fact that even in a non-distributive lattice, there still holds a distributive inequality. These calculations can easily be visualized by Venn diagrams. Each element f L corresponds to a set in R2, and the valuation v equals the area of this set. ∈ The distance between f and g then can be seen as the area of the symmetric difference of f and g. Now the union of the symmetric differences of f and h on the one hand, and g and h on the other hand, is a superset of the symmetric difference of f and g. Hence, d (f, g) d (f, h) + d (h, g). v ≤ v v However, this metaphor is not a full proof, as the operation of symmetric difference fails for general lattices. We will return to this matter in Lemma 64 in a more general context. 30 Claim 55

g x g x g x g x

f y f y f y f y d(f v g, x v y) + d(f v g, x v y) d(f, x)+ d(g, y)

Figure 1.4: Another property of valuation metrics, visualized with a four-set Venn diagram; a precursor to Proposition 72.

Example 54 ...... 54 We continue with the special case L = (Z, min, max) from example 50. Although it was not possible to create a link between the valuation v and the counting measure µ, the diﬀerence valuation w on L provides us with a useful connection:

w(n,m) = µ(A(n) r A(m)) In cases of distributive lattices without least elements, it is feasible and beneficial to forget about v and define dv directly in terms of the difference valuation. Proposition 51 shows that this is possible without loss of information.

We want to demonstrate the use of Venn diagrams to derive calculation rules in valuation metric lattices:

Proposition 55 ...... 55 Let (L, , ) be a distributive lattice with diﬀerence valuation w and corre- ∧ ∨ sponding valuation metric d. Then for all f, g, x, y L holds: ∈ d(f g, x y) + d(f g, x y) d(f, x) + d(f, y) ∧ ∧ ∨ ∨ ≤ Proof Figure 1.4 depicts the proposition. As the symmetry of the ﬁgure suggests, we can prove the slightly stronger statement

w(f g, x y) + w(f g, x y) w(f, x) + w(f, y) (*) ∧ ∧ ∨ ∨ ≤ from which the thesis directly follows. Next, we decompose each term into smaller subterms with the help of the cut and absorption laws, to gain subterms of the form w(a . . . a , b . . . b ) in which each of f, g, x, y appear at 1 ∧ ∧ n 1 ∨ ∨ n exactly one position. These are minimal terms, which each correspond to one of the ﬁfteen minimal areas in the Venn diagram. Figure 1.4 helps us to determine along which function we have to cut. We get:

w(f g, x y) = w(f g x, y) + w(f g y, x) + w(f g, x y) ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∨ w(f g, x y) = w(f g, x y) + w(f, g x y) + w(g, f x y) ∨ ∨ ∧ ∨ ∨ ∨ ∨ ∨ w(f, x) = w(f g y, x) + w(f y, g x) ∧ ∧ ∧ ∨ + w(f g, x y) + w(f, g x y) ∧ ∨ ∨ ∨ w(g, y) = w(f g x, y) + w(g x, f y) ∧ ∧ ∧ ∨ + w(f g, x y) + w(g, f x y) ∧ ∨ ∨ ∨ Valuation Metric Lattices 31

It is possible to recombine the remaining subterms and express the diﬀerence of left- and right-hand side as another sum of distances—we leave this as recre- ation to the reader.

Example 56 ...... 56 If L = Lip X, with X a measure space, we may apply the Lebesgue integral to gain an isotone valuation on L; as f + g = (f g) + (f g) holds pointwise, we ∧ ∨ conclude

f dµ + g dµ = (f g)dµ + (f g)dµ. ∧ ∨ Z Z Z Z If X is a Euclidean space, or a discrete space without non-trivial null sets, this valuation is positive, because any non-trivial non-negative Lipschitz function has positive Lebesgue integral. Positivity fails in cases where X contains an isolated point or continuum of measure zero. As f g = (f g) (f g) holds pointwise, the valuation metric d equals | − | ∨ − ∧ v the L1-distance deﬁned by

d (f, g) := f g dµ. 1 | − | Z This raises the question for which metrics d on Lip X there is a valuation v, such that d = dv. From property (2) in Lemma 53 we can easily deduce the valuation v. Insertion into the deﬁnition of a valuation leads to the requirement d (f, 0) + d (g, 0) = d (f g, 0) + d (f g, 0) f, g Lip X. v v v ∨ v ∧ ∀ ∈ In the special case of functions f, g with disjoint support we have f g = f + g, ∨ so we end up with the necessary condition f g = 0 d (f, 0) + d (g, 0) = d (f + g, 0). ∧ ⇒ v v v Now it should not take us any wonder that “completely non-linear” metrics like the supremum metric or

d (f, g) := p f g p dµ p | − | sZ for p> 1 are non-valuation metrics.

Example 57 ...... 57 Yet, there are some more valuation metrics besides the L1-metric. Let κ : [0, ) [0, ) be a positive valuation (i.e., strictly monotonically increasing), ∞ → ∞ then

vµ,κ(f) := κ(f(x)) dµ(x) Z is a positive valuation. However, the author is not aware of any valuation metric on Lip X, which cannot be described in this way with suitable κ and µ. 32 Claim 60

1.2 Ultravaluation Metrics

Lemma 58 ...... 58 Let L be a distributive lattice, and let w : L L [0, ] be a map which × → ∞ satisﬁes

(1) w(f, g) = 0 whenever f g, and ≤ (2) w(f, g) = w(f h, g) w(f, g h) f,g,h L. ∧ ∨ ∨ ∀ ∈

We call w a diﬀerence ultravaluation, or just ultravaluation. Deﬁne dw(f, g) := w(f, g) w(g,f). Then d is a pseudo-ultrametric . d is an ultrametric if ∨ w + w + and only if w(f, g) = 0 f g holds. ⇒ ≤

Proof To get from normal valuations to ultravaluations, we just replaced all occurences of “+” by “ ”. As both operations are associative and commutative, ∨ we can transfer most proofs of valuations just by replacing “+” by “ ”: ∨ d (f, g) = w(f, g h) w(f h, g) w(g,f h) w(g h, f) w ∨ ∨ ∧ ∨ ∨ ∨ ∧ w(f, g h) w(f, h) etc. ∨ ≤ d (f, g) w(f, h) w(h, g) w(g, h) w(h, f) = d (f, h) d (h, g) ⇒ w ≤ ∨ ∨ ∨ w ∨ w

On the other hand, contrary to the valuation case, the property dv(f,f) = 0 does not follow from property (2) – we have to conclude it from (1). Assume w(f, g) = 0 f g holds. Let d (f, g) = 0. This implies w(f, g) = 0 ⇒ ≤ w and w(g,f) = 0, and hence f g, g f, and f = g. Now assume d is a ≤ ≤ w metric, f g, and w(f, g) =0. Then

w(f,f g) = w(f g,f g) w(f, g) = 0. ∧ ∧ ∧ ∨ Due to f g, we have f = f g, hence 6 ∧ 0 < d (f,f g) = w(f,f g) w(f g,f) = w(f g,f). w ∧ ∧ ∨ ∧ ∧ But f g f, contradiction. ∧ ≤

Example 59 ...... 59 Let X be any set, κ : X [0, ] arbitrary and ﬁxed, and L a lattice of subsets → ∞ of X. For A, B L consider ∈ w(A, B) := 0 sup κ(x). ∨ x ArB ∈ w deﬁnes an ultravaluation. Choose κ to be a positive constant, then the ultrametric resulting from w will be the discrete metric on X. Ultravaluation Metrics 33

Lemma 60 ...... 60 Let X be ﬁnite, and let L be a lattice of subsets of X. Then any ultravaluation on L is of the form of Example 59.

Proof For x X and A, B X deﬁne ∈ ⊆ κ(x) := inf w(C, D) : C, D L with x C, x / D { ∈ ∈ ∈ } and w′(A, B) := 0 sup κ(y). ∨ y ArB ∈ Assume w (A, B) > w(A, B). Then there is y A r B with κ(y) w(A, B), ′ ∈ ≥ but this cannot happen, as one may choose C = A and D = B. Hence, assume w (A, B) < w(A, B). Then for all y A r B there should be C, D L with ′ ∈ ∈ y C r D and w(C, D) < w(A, B). As ∈ w(C, D) w(C A, D B), ≥ ∧ ∨ we might choose without loss of generality C A and D B, as choosing ⊆ ⊇ C A instead of C and D B instead of D further decreases w(C, D). The cut ∩ ∪ law now yields

w(A, B) = w(C D, B) w(C, D) w(A D, B C) w(A, C D). ∧ ∨ ∨ ∨ ∨ ∨ ∨ As w(C, D) < w(A, B) by assumption, we find that at least one of (C D)rB, ∩ (A D) r (B C), and A r (C D) must be non-empty. Choose y out of their ∪ ∪ ∪ ′ union and repeat the above argument for the now smaller subset. We get an infinite sequence of different elements from X, which is a contradiction because X is finite.

Example 61 ...... 61 Now consider L = Lip X, with X some metric+ space. We represent L with subsets of X [0, ], as described in Example 45, to apply it to Example 59. The × ∞ most canonical κ would be κ = π , the projection onto [0, ]. The corresponding 2 ∞ ultrametric on L is

d (f, g) = 0 sup f(x) g(x) with x X such that f(x) = g(x) . κ ∨ { ∨ ∈ 6 } We shall call this metric the “peak metric” on Lip X. Another possible choice for κ is as follows: Choose a basepoint x X and 0 ∈ κ(x, r) := dX (x,x0). Then dκ will describe the greatest distance from x0 at which f and g still differ. Finally, κ(x, r) := exp( d (x,x )) will describe − X 0 the least distance from x0 at which f and g differ. We will call the first case the “outer basepoint metric” and the second case the “inner basepoint metric”. An application of the lower basepoint metric is as follows: Given a free group F with neutral element x0, identify each normal subgroup N E F with its characteristic function on F . These are 1-Lipschitz functions in the canonical word metric of F . dκ then defines a topology on Lip F , which restricts to the Cayley topology ([dH], V.10) on the subset of normal subgroups. 34 Claim 63

1.3 Intervaluation Metrics

We now integrate the supremum metric into the context of difference valuation, but not without a sincere generalization of the concept. Similar to the case of the ultravaluation, We first recognize the possibility to replace “+” in the definition of a difference valuation by any commutative and associative binary operation. But this alone will not suffice to encompass the supremum metric, we have to weaken the main property of a difference evaluation as well:

Deﬁnition 62 ...... 62 An intervaluation on a distributive lattice (L, , ) is a map w : L [0, ] ∧ ∨ → ∞ together with a commutative and associative binary operation : [0, ] ◦w ∞ × [0, ] [0, ], such that the following properties hold: ∞ → ∞ 1. r 0 = 0 r = r ◦w ◦w 2. r t (r + s) (t + u) (r t) + (s u) ◦w ≤ ◦w ≤ ◦w ◦w 3. r s r s (follows from (1) and (2)) ∨ ≤ ◦w 4. f g w(f, g) = 0 ≤ ⇒ 5. w(f, g h) w(f h, g) w(f, g) w(f, g h) + w(f h, g) ∨ ◦w ∧ ≤ ≤ ∨ ∧ (left and right modular inequality, or cut law) for all f,g,h L and r,s,t,u [0, ]. The corresponding intervaluation metric ∈ ∈ ∞ then is deﬁned to be

d (f, g) := w(f, g) w(g, f). w ◦w The intervaluation is positive if

w(f, g) = 0 f g. ⇒ ≤

Proposition 63 ...... 63 An intervaluation w on L and its metric dw always fulﬁll:

1. w(f, g) = w(f g, g) = w(f, f g) = d (f g, g) f, g L. ∨ ∧ w ∨ ∀ ∈

2. dw is a pseudo-metric+ .

3. dw is a metric+ if and only if w is positive.

Proof (1) We choose h = f or h = g in both modular inequalities:

0 w(f, g) w(f g, g) 0 + w(f, g) ◦w ≤ ∨ ≤ w(f, g) 0 w(f, g f) w(f, g) + 0 ◦w ≤ ∧ ≤ and d (f g, g) = w(f g, g) 0 = w(f, g). w ∨ ∨ ◦w Intervaluation Metrics 35

(2) From the deﬁnition we see d (f, g) 0 and d (f, f) = 0 for all f, g L. w ≥ w ∈ As is commutative, d is symmetric. ◦w w d (f, g) = w(f, g) w(g, f) w ◦w (w (f h, g) + w (f, g h)) (w (g h, f) + w (g, f h)) ≤ ∧ ∨ ◦w ∧ ∨ (w (h, g) + w (f, h)) (w (h, f) + w (g, h))) ≤ ◦w = (w (f, h) + w (h, g)) (w (h, f) + w (g, h))) ◦w (w (f, h) w (h, f)) + (w (h, g) w (g, h))) ≤ ◦w ◦w = dw(f, h) + dw(h, g)

(3, “ ”) Assume 0 = w(f, g) = w(f, f g). Then d (f,f g)=0+0=0. ⇒ ∧ w ∧ As d is a metric, we have f = f g, so f g. w ∧ ≤ (3, “ ”) d (f, g) = 0 implies w(f, g)=0 and w(g, f) = 0, hence f g f, ⇐ w ≤ ≤ and f = g.

The definition of intervaluations is chosen to generalize difference valuations and ultravaluations, while maintaining as many inequalities as possible. As those calculation laws derived from Venn-diagrams hold for difference valuations and ultravaluations likewise, which both work as the extremal cases of intervaluations, it is a worthwhile endeavour to figure out those laws that still hold for intervaluation metrics, which contain lots of interesting metrics for Lipschitz function spaces.

Lemma 64 ...... 64 Let L be a distributive lattice with intervaluation metric+ d. Define a polynomial in L as an expression of finitely many variables in L, linked by and . ∨ ∧ Define a difference term in L to be a term of the form w(x, y), where x and y are polynomials in L. Define a difference polynomial in L as a term built from finitely many difference terms, linked by addition. Let and be difference T1 T2 terms. Let f to f be the set of all variables occuring in and . Draw a 1 N T1 T2 Venn diagram with f1 to fN as basis, and let π be the correspondence between the subsets of the Venn diagram and polynomials of f1 to fN in L. Define ρ(x, y) := π(x) r π(y). Define π( ) to be the union of all ρ(x, y) for which T1 w(x, y) appears in . If π( ) π(T ), then (2N 1) . T1 T1 ⊆ 2 T1 ≤ − ·T2

Proof Call the 2N 1 minimal subsets in the Venn diagram atoms. Each atom − A is uniquely described by a non-empty subset S of f ,..., f , such that { 1 N }

A = π(f ) r π(f ) ,  j   j  fj S fj / S \∈ [∈     and hence A = ρ( S fj, Sc fj). On the other hand, each diﬀerence term w(x, y) can be uniquely chopped down into corresponding atoms of the form V W w( S fj, Sc fj) as well, by repeatedly using the cut law, up to N times (for V W 36 Claim 65 each variable once; as and distribute over each other as well as over them- ∧ ∨ selves, the absorption law reduces each polynomial to a polynomial of or ∧ ∨ only); uniqueness follows from commutativity. By Deﬁnition 62 we conclude: maximum of up to (2N 1) atoms w(x, y) sum of up to (2N 1) atoms − ≤ ≤ −

As j J aj #J maxj J aj for any ﬁnite index set J and real numbers aj, ≤ · ∈ we have∈ P sum of atoms (2N 1) w(x, y) (2N 1) (sum of atoms) ≤ − · ≤ − · Correspondingly, each diﬀerence term can be bounded from above and below T by multiples of the atoms of its constituents. If π( ) π( ), then each atom T1 ⊆ T2 of the left-hand Venn diagram occurs in the right-hand Venn diagram as well, hence the sum of atoms in is a subsum of the sum of atoms in . We then T1 T2 follow sum of -atoms sum of -atoms (2N 1) . T1 ≤ T1 ≤ T2 ≤ − ·T2

Example 65 ...... 65 There are several possible choices for the commutative and associative binary operation in Definition 62. Choosing addition leads directly to the definition ◦w of valuations. The next important choice is the maximum operation: Properties (1) and (3) are obviously fulfilled, the left side of (2) as well. (2.right) needs some short consideration: As + distributes over , the right-hand side equals ∨ (r t) + (s u) = (r + s) (r + u) (t + s) (t + u) ∨ ∨ ∨ ∨ ∨ which is greater or equal (r + s) (t + u) for all r, s, t, u [0, ]. ∨ ∈ ∞ Each norm on R2 with certain normalization properties qualifies as an || · || operation via r s := (r, s) . This accounts for the p-norms: ◦w ◦w || || r s := (r, s) := √p rp + sp ◦p p for p [1, ). Again, properties (1), (2.left) and (3) of Definition 62 are triv- ∈ ∞ ial. Property (2.right) is the triangle inequality of the p-norms (i.e. a special case of the Minkowski inequality [Wr]).

Given any metric d on L we may deﬁne w (f, g) := d(f g, g) and deduce d ∨ ◦w from d(f, g) = w (f, g) w (g, f). The operation must be commutative d ◦w d ◦w due to the symmetry of dw. From the remaining properties of Deﬁnition 62, property (4) follows directly from d(g, g) = 0, while the rest is less obvious.

Example 66 ...... 66 The standard metric on [0, ] is an intervaluation metric with + ∞ + 0 (r s) : s < w(r, s) := ∨ − ∞ r, s [0, ]. 0 : s = ∀ ∈ ∞ ∞ Intervaluation Metrics 37

However, one may freely choose to be addition or maximum. To prove the ◦w cut law for both choices, it suﬃces to show

0 (r s) = 0 r (s t) + 0 (r t) s) . ∨ − ∨ − ∨ ∨ ∧ − For this, we make use of a+ b = (a b) + (a b) with a = r s and b = r t, ∧ ∨ ∧ ∧ then add r to both sides, rearrange and apply x (x y) = 0 (x y). − ∧ ∨ − Example 67 ...... 67 Let (X, µ) be a measure space, L = Lip X, and p (1, ) arbitrary. Deﬁne ∈ ∞ r s = (rp + sp)1/p, and ◦w

w(f, g) := p f (f g) p dµ . − ∧ sZ

As r (r s) p + s (r s) p = r s p for all r, s [0, ] (with p := ), | − ∧ | | − ∧ | | − | p ∈ ∞ ∞ ∞ the corresponding (pseudo-)metric+ is just the L -metric+

d (f, g) = p f g p dµ . p | − | sZ Properties (1)-(3) of Deﬁnition 62 follow from Example 65, (4) is trivial. The left cut law can be shown by pointwise analysis and case distinction (h g vs. ≤ h > g), the right cut law follows from Example 66 and the Minkowski inequality. dp might be a pseudo-metric+ , depending on µ.

Example 68 ...... 68 Here is a minimal example for a non-intervaluation metric: Take L = a, b, c { } with a

LC(f g) LC(f g) LC(f) LC(g) ∨ ∨ ∧ ≤ ∨ 38 Claim 71 the inverse inequality is wrong, as there is no bound to LC(f) by any combi- nation of LC(f g) and LC(f g). To see this, consider the two-point-space ∧ ∨ X = a, b of diameter l < 1, and the Lipschitz-functions f = (0, l) and { } g = (l, 0). Then LC(f) = f = 1, but LC(f g) = LC(f g) = 0 and || ||L ∧ ∨ = l in both cases. || · ||L Correspondingly, the cut law is explicitly violated by dLC, as one can see when f and g are two diﬀerent constant functions, and h crosses them both.

We now concentrate on the special case of the supremum metric.

Proposition 70 ...... 70 Let Z be a distributive lattice with intervaluation metric+ d (with corresponding w and ), with r s = r s for all r, s [0, ]. Let X be an arbitrary d ◦d ◦d ∨ ∈ ∞ space, and L a complete lattice of functions f : X Z with pointwise inﬁma → and suprema. Then

w (f, g) := wd f(x), g(x) ∞ x X _∈ deﬁnes an intervaluation metric+ on L with r s = r s for all r, s [0, ], ◦∞ ∨ ∈ ∞ which equals the supremum metric+ d . ∞ Proof The left inequality of the cut law is trivial. For the right side we have to use that a supremum of sums is less than or equal to a sum of suprema, which in turn follows from complete distributivity:

w fx, gx w fx, (g h)(x) + w (f h)(x), gx d ≤ d ∨ d ∧ x X x X _∈ _∈ w fx, (g h)(x) + w (f h)(x), gx ≤ d ∨ d ∧ x X x X _∈ _∈

Corollary 71 ...... 71 Let X be any metric+ space. The supremum metric+ d is an intervaluation ∞ metric+ on Lip X.

Proof We have seen in Proposition 43 that Lip X is a complete lattice. We easily ﬁnd r s = r s and ◦d∞ ∨ w (f, g) = f(x) (f g)(x) = 0 f(x) g(x) , d∞ − ∧ ∨ − x X x X _∈ _∈ which is the intervaluation metric+ of Proposition 70 applied to Example 66.

Apart from all laws which we may derive from Venn diagrams, the supremum metric on Lip X provides yet another interesting law, which is an adaptation and generalization of Proposition 55: Complete Metric/Lattice-Irreducibility 39

Proposition 72 ...... 72 For f , g arbitrary set theoretic functions X [0, ], j J, J some arbitrary j j → ∞ ∈ index set, holds:

d fj, gj d (fj, gj) ∞   ≤ ∞ j J j J j J ^∈ ^∈ _∈  

d fj, gj d (fj, gj) ∞   ≤ ∞ j J j J j J _∈ _∈ _∈  

Proof For J = , both inequalities are trivial. Assume J = . As j J and ∅ 6 ∅ ∈ x X commute, it suffices to show ∈ W W d xj, yj d (xj,yj) ∞ ≤ ∞ ^ ^ _ d xj, yj d (xj,yj) ∞ ≤ ∞ _ _ _ for any x ,y [0, ]. j j ∈ ∞ First we handle infinities. First inequality: Assume there is j with x = y = . j j ∞ We can ignore all such j’s from J, unless all x and y are . In this case on j j ∞ both sides are zeros. Now assume x = = y . Then appears on the right j ∞ 6 j ∞ side and trivializes the inequality. So we can restrict to finite xj and yj. Note that x = can only happen when all x = . j j ∞ j ∞ Second inequality: Assume x = , but y is finite. Then there is an V j j ∞ j j upper bound for y but not for x . Hence the right side becomes infinite, j W j W too. Note that infinite xj or yj automatically lead to infinite j xj or j yj, respectively. W W Without restriction let x y , and let M := d(x ,y ). Let δ > 0 be j j ≥ j j j j j arbitrary. Then there is an m J with y y + δ. Furthermore we have V ∈V m ≤ j j W d(x ,y ) M, hence y x M. Altogether: m m ≤ m ≥ m − V x x y + M + δ j ≤ m ≤ j Now let δ 0. The other^ inequality works^ the same way. →

1.4 Complete Metric/Lattice-Irreducibility

Recall Deﬁnition 36 of a join-irreducible element p in a lattice L:

p = f g p = f or p = g f, g L ∨ ⇒ ∀ ∈ Let L be equipped with the discrete metric ddis. Then the above property is equivalent to the following:

d (p, f) d (p, g) d (p, f g) f, g L dis ∧ dis ≤ dis ∨ ∀ ∈ 40 Claim 75

In the same sense, p is completely irreducible if and only if

ddis p, fj ddis p, fj (fj)j J L, J = . ≤   ∀ ∈ ⊆ 6 ∅ j J j J ^∈ _∈   Deﬁnition 73 ...... 73 Let L be a complete lattice with intervaluation metric d. We call p L com- + ∈ pletely ml-irreducible if the following equivalent conditions hold:

1. (fj)j J L, with J an arbitrary non-empty index set: ∀ ∈ ⊆

d p, f d p, f j ≤  j j J j J ^∈ _∈  

2. (fj)j J L, J an arbitrary non-empty index set, and R [0, ] : ∀ ∈ ⊆ ∀ ∈ ∞

d p, f R δ > 0 j J : d (p,f ) R + δ  j ≤ ⇒ ∀ ∃ ∈ j ≤ j J _∈   (This results from expanding the inﬁmum in (1).)

Let 0 L be the least element in L, then a bounded completely ml-irreducible ∈ element p L is a completely ml-irreducible element with ﬁnite distance to 0. ∈ Denote the subset of L of all [bounded] completely ml-irreducible elements with cmli(L) [bcmli(L)].

Proposition 74 ...... 74 Let L be metrically closed, then each completely ml-irreducible element is join- irreducible. It is not necessarily completely join-irreducible (see Deﬁnition 36).

Proof Use R = 0 and that a set of two elements is compact. For a counter- example to complete join-irreducibility, let L = [0, 1] with standard metric, supremum and inﬁmum. Take f = 1 1/n, n N , then p =1 = f , n − ∈ ∗ n hence p is not completely join-irreducible. Still, it is completely ml-irreducible: W Any sequence of real numbers fn with p = fn must converge to p from below, hence d(p, f ) = 0. n W V

Proposition 75 ...... 75 Let L be a complete lattice with intervaluation metric+ d, and let L be metrically complete. Then cmli(L) is topologically closed. If 0 is the least element of L, then bcmli(L) is topologically closed, and 0 bcmli(L). ∈ Complete Metric/Lattice-Irreducibility 41

Proof Let (p ) cmli(L), n N be some sequence of completely ml-irre- n ⊆ ∈ ∗ ducible elements converging to p L, and (fj)j J any non-empty family in L. ∈ ∈ Then for any n N holds ∈ ∗

d p, f d p , f d(p, p ) j ≥ n j − n _ d(p _, f ) d(p, p ) ≥ n j − n ^ d(p, f ) d(p, p ) d(p, p ) ≥ j − n − n ^ d(p, f ) 2 d(p, p ), ≥ j − n 0 ^ → | {z } i.e. the element p is completely ml-irreducible. Any component of L is topologically closed, in particular the component of 0, hence the intersection with cmli(L) is closed as well. Furthermore, we have for any k J = : ∈ 6 ∅

d 0, fj = wd fj, 0 idempotency _ = w f_ f , 0 cut along f d k ∨ j k w (f , 0) _ w f ,f ≥ d k ◦w d j k w (f , 0) = d(0_, f ) ≥ d k k and thus d(0,f ) d(0, f ) d 0, f . j ≤ k ≤ j V W

Deﬁnition 76 ...... 76 Let L be a lattice with metric d, R 0 arbitrary. We deﬁne an R-base of + ≥ L to be a subset B L such that for any f L there is (bj)j J B, J an ⊆ ∈ ∈ ⊆ arbitrary non-empty index set, such that d(f, b ) R.A base simply j J j ≤ is a 0-base. Given a least element 0 L, a bounded∈ R-base is an R-base of L ∈ W which is a subset of the component of 0.

Example 77 ...... 77 Let L = [0, ] with standard metric d (modulus of the diﬀerence). Then one ∞ + easily calculates cmli(L) = bcmli(L) = [0, ). Furthermore, any metrically ∞ dense subset B of [0, ) is a bounded base, in particular, we have sequences ∞ (b ) B with b = , although d(b , ) = , and thus b in d. j ⊆ j ∞ j ∞ ∞ j 6→ ∞ W Proposition 78 ...... 78 Consider an R-base B of a complete lattice L with intervaluation metric+ d, R 0. Then for each δ > 0, cmli(L) is in the (R + δ)-ball around B. In ≥ particular, if R = 0, cmli(L) lies in the metrical closure of B. 42 Claim 79

Proof Let p cmli(L) be arbitrary. As B is an R-base, there are b B, ∈ j ∈ j J = , such that ∈ 6 ∅

d p, b R.  j ≤ j J _∈   From Deﬁnition 73 we infer that there is a sequence (c ) B, k K J k ⊆ ∈ ⊆ whose distances to p converge to R. If R = 0, the sequence (cj) metrically converges to p.

Example 79 ...... 79 It is easy to see that, if B is a base, and b B not a join-irreducible element, ∈ then B r b is a base as well (if b = f g, f and g are joins of elements of { } ∨ B, and as f,g < b, b is not part of these joins). Using the Lemma of Zorn, it is possible to deduce that the subset of all join-irreducible elements constitutes a base for any suﬃciently nice lattice.

Unfortunately, this is not the case with ml-irreducible elements: Let L′ be the completely distributive complete lattice [0, 3] [0, 2] with componentwise supre- × mum and infimum, and with supremum metric. Then consider the sublattice L L formed by the five elements ⊆ ′ L := (0, 0), (1, 0), (0, 1), (1, 1), (2, 2) . { } We find cmli(L) = (0, 0), (1, 0), (0, 1) , as (1, 1) = (1, 0) (0, 1). p = { } ∨ (2, 2) is join-irreducible in this lattice, but not ml-irreducible: Take f1 = (1, 0), f2 = (0, 1), then d(p, fj) = 2, but d(p, fj) = 1. Nevertheless, (2, 2) must be part of any 0-base of L. V W Chapter 2

Rough Isometries of Lipschitz Function Spaces

2.1 Smoothening of Lipschitz Functions

Definition 80 ...... 80 Let X and Z be metric spaces, and ǫ, K 0 fixed. We define the Z-valued + ≥ K-Lipschitz and (K,ǫ)-Lipschitz functions on X by:

Lip (X, Z) := f : X Z : x,y X : d(fx,fy) K d(x,y) K { → ∀ ∈ ≤ } Lip (X, Z) := f : X Z : x,y X : d(fx,fy) K d(x,y)+ ǫ K,ǫ { → ∀ ∈ ≤ } Later on we will restrict to Z = R and Z = [0, ]. ∞ Given two metric spaces X, Y and a (λ, ǫ)-quasi-isometry η : X Y , we can + → push η to a map between the Lipschitz function spaces

η∗ : LipK,δ(Y, Z) LipK λ, K ǫ+δ(X, Z) → · · f f η. 7→ ◦ Using the supremum metric (aka. L∞-metric) d on Lipschitz functions, we ∞ easily see that η∗ is a (2Kǫ + 2δ)-isometric embedding of LipK,δ(Y, Z) into LipK λ, K ǫ+δ(X, Z): Obviously, we have · · d(fηx,gηx) d (f, g) x X. ≤ ∞ ∀ ∈ On the other hand, for each 0 D < d (f, g) we may ﬁnd y Y with ≤ ∞ ∈ d(fy,gy) D, and x X with d(ηx,y) ǫ. Hence, ≥ ∈ ≤ d(fηx,gηx) d(fy,gy) d(fηx,fy) d(gηx,gy) ≥ − − D 2 Kǫ 2 δ. ≥ − − Taking the supremum over x, we conclude d (η∗f, η∗g) d (f, g) 2Kǫ 2δ. ∞ ≥ ∞ − −

Question Is η∗ even a quasi-isometry? 44 Claim 81

We may split this problem into two special cases: The rough case with λ = 1. And the bilipschitz case ǫ = 0, for which we can give an easy counterexample: Let X be R2 with Euclidean metric, and let Y be R2 with supremum metric. The identity is a 2-bilipschitz bijection. Let Z = R and f : X Z, (x ,x ) → 1 2 7→ x2 + x2, i.e. f(x)= x , and f Lip (X, Z). In particular, for points x with 1 2 | |X ∈ 1 x = x we find f(x)= √2 x . But for g Lip (Y, Z) must hold p1 2 · 1 ∈ 1 g(x) g(0) sup x ,x , | − | ≤ { 1 2} such that with x , f and g have infinite distance from each other. One 1 → ∞ would argue that in this case, we have to choose g Lip (Y, Z). But then, ∈ √2 we may choose points x = (x , 0) to see that the function (x ,x ) √2 x 1 1 2 7→ · | |Y has infinite distance to any f Lip (X, Z). To solve this problem, one has to ∈ 1 generalize Lipschitz functions to allow for varying Lipschitz constants – however, our main focus lies on the rough isometry case λ = 1. As

Lip (X, Z) Lip (X, Z) K ⊆ K,ǫ for any ǫ 0, we may reformulate our question in the following way: ≥ Question

How L∞-dense is the subset of all K-Lipschitz functions in the metric+ space of (K, ǫ)-Lipschitz functions?

We will next give an answer to this problem in the special case Z = R. A similar result for continuous functions is given by Petersen in [P], section 4.

Lemma 81 ...... 81 Let X be a separable metric space, K,ǫ > 0 and let f : X R be a function + → with

fx fy K d(x,y) + ǫ. | − | ≤ · Then there is a K-Lipschitz function g : X R with → d (f, g) 2 ǫ. ∞ ≤

Proof Let f : X R bea(K,ǫ)-Lipschitz function, K,ǫ 0. Let aj j N∗ → ≥ { } ∈ ⊆ X be a countable dense subset. Deﬁne g(a1) := f(a1). Now deﬁne inductively and prove by induction the following:

• j 1 − I := g(a ) K d(a , a ), g(a )+ K d(a , a ) j k − · j k k · j k k=1 \ Smoothening of Lipschitz Functions 45

• f(a ) : f(a ) I j j ∈ j g(a ) := min I : f(a ) min I j  j j ≤ j max I : f(a ) max I  j j ≥ j  (a): g a1,..., aj is K-Lipschitz. • |{ } (b): f(a ) g(a ) ǫ. • | j − j |≤ We see that (a) and (b) are trivially fulﬁlled for j = 1. At ﬁrst we have to show that Ij = : As g a1,..., aj−1 is K-Lipschitz, we know for all m,n

g(a ) g(a ) K d(a , a ) k < j | j − k |≤ · j k ∀ and together with the Lipschitz property of g on a1,..., aj 1 we see that g { − } is also K-Lipschitz on a ,..., a . { 1 j} Finally, we show f(a ) g(a ) ǫ. If f(a ) I , we have nothing to show. | j − j | ≤ j ∈ j Assume the second case: f(a ) min I and g(a ) = min I . Let n be such that j ≤ j j j f(a ) g(a ) = min I = g(a ) K d(a , a ) j ≤ j j n − · n j

By deﬁnition of Ij we have

g(a ) K d(a , a ) g(a ) K d(a , a ) m < j n − · n j ≥ m − · m j ∀ for some n, and we choose n to be the least possible index with this property. Furthermore, we have

f(a ) f(a ) K d(a , a )+ ǫ | j − n | ≤ · j n f(a ) f(a ) K d(a , a ) ǫ. ⇒ j ≥ n − · j n − We use this to calculate

0 g(a ) f(a ) = g(a ) K d(a , a ) f(a ) ≤ j − j n − · n j − j g(a ) K d(a , a ) f(a )+ K d(a , a )+ ǫ ≤ n − · n j − n · n j g(a ) f(a )+ ǫ. ≤ n − n

Now assume g(an) >f(an). Then there is some m

g(a ) = g(a ) K d(a , a ) n m − · n m g(a ) K d(a , a ) = g(a ) K d(a , a ) K d(a , a ) ⇒ n − · n j m − · n m − · n j g(a ) K d(a , a ) ≤ m − · m j 46 Claim 82 which contradicts the minimality of n. Hence we have g(a ) f(a ) and n ≤ n g(a ) f(a ) ǫ. | j − j |≤ The third case (f(aj) > g(aj) = max Ij) works the same:

g(a ) = g(a )+ K d(a , a ) n smallest possible j n · n j 0 f(a ) g(a ) = f(a ) g(a ) K d(a , a ) ≤ j − j j − n − · n j f(a ) f(a )+ K d(a , a )+ ǫ | j ≤ n · n j f(a ) g(a )+ ǫ ≤ n − n Assume f(a ) > g(a ), then m

Theorem 82 82 Let X, Y be separable metric spaces, and η : X Y , ξ : Y X both ǫ- + → → isometries (ǫ 0) with η ξ and ξ η being 2 ǫ-near to id respectively id . ≥ ◦ ◦ Y X Then for each K 0 the spaces of K-Lipschitz-functions are 6 Kǫ-isometric, ≥ in particular there exists a 6 Kǫ-isometry

η¯ : Lip (Y ) Lip (X) K → K in respect to the L∞-metric on LipK, with d (¯η(f), f η) 2 Kǫ ∞ ◦ ≤ and a corresponding ξ¯.

Proof Choose dense countable subsets a X and b Y . For each { j} ⊆ { j} ⊆ f Lip (Y ) we have f η : X R which fulﬁlls ∈ K ◦ → d(fηx,fηy) K d(ηx,ηy) K d(x,y)+ K ǫ. ≤ ≤ · Smoothening of Lipschitz Functions 47

We smooth this function in the way of Lemma 81 with respect to a (note { j} that ǫ is replaced by Kǫ) and get a K-Lipschitz-function which we call

η¯(f) : X R → with d (¯η(f),f η) 2 Kǫ. We now show that the mapping ∞ ◦ ≤ η¯ : Lip (Y ) Lip (X) K → K is a rough isometry with respect to d . For this, we ﬁrst take a look at ∞ d η¯(f), η¯(g) 2 Kǫ + d (f η, g η) + 2 K ǫ. ∞ ≤ ∞ ◦ ◦ for arbitrary f, g Lip (Y ). It is clear that ∈ K d (f η, g η) d (f, g) ∞ ◦ ◦ ≤ ∞ d η¯(f), η¯(g) 4 Kǫ + d (f, g). ⇒ ∞ ≤ ∞ On the other hand we have for each y Y an x X with d(ηx,y) ǫ, and ∈ ∈ ≤ therefore for all y Y : ∈ d(fy,gy) d(fy,fηx)+ d(fηx,gηx)+ d(gηx,gy) ≤ 2 Kǫ + d(fηx,gηx) ≤ hence

d (f, g) 2 Kǫ + d (fη,gη) ∞ ≤ ∞ 6 Kǫ + d (¯η(f), η¯(g)), ≤ ∞ which proves the ﬁrst part of a rough isometry. In the same way we deﬁne ξ¯ : Lip (X) Lip (Y ). We now have K → K d η¯(ξf¯ ),f d η¯(ξf¯ ), [ξf¯ ] η + d [ξf¯ ] η,f ξ η ∞ ≤ ∞ ◦ ∞ ◦ ◦ ◦ 2 Kǫ + 2 Kǫ + 2 K ǫ. ≤ Note that for the second term we made use of

d [ξf¯ ] η,f ξ η d (ξf,f¯ ξ) 2 K ǫ. ∞ ◦ ◦ ◦ ≤ ∞ ◦ ≤ In the same way we approximate η ξ and ﬁnd that they both are 6 Kǫ-near ◦ to the corresponding identities on LipK(X) and LipK(Y ). This last step also proves the second property of rough isometries: For f Lip (X) ∈ K choose ξ¯(f) and conclude d(¯η(ξf¯ ),f) 6 Kǫ. ≤ 48 Claim 83

2.1.1 Hyperconvex and Non-Hyperconvex Codomains

Unfortunately, the two previous proofs are only slightly generalizable to other choices for Z, in particular for hyperconvex metric+ spaces.

Lemma 83 ...... 83 Let X be a separable metric space, Z a hyperconvex metric space, and ǫ, K + + ≥ 0 arbitrary. Then LipK (X, Z) is 2ǫ-dense in LipK,ǫ (X, Z).

Proof Let f Lip (X, Z) and x X be arbitrary, and choose a countable ∈ K,ǫ ∈ dense subset aj j N∗ X. By induction, deﬁne g(aj ) as follows: Deﬁne { } ∈ ⊆ r := K d(a , a ) and s := r + ǫ for any k N and k · k j k k ∈ ∗

:= B (r ) : 1 k < j B (s ) : k N∗ . Fj g(ak) k ≤ ∪ f(ak) k ∈ We show that hyperconvexity applies and freely choose g(a ) : j ∈ Fj T 1. d g(a ), g(a ) r + r . WLOG assume n

2. d f(a ), g(f ) s + s . This holds from the fact that f is (K, ǫ)- n m ≤ n m Lipschitz, plus triangle inequality. 3. d f(a ), g(a ) s + r . Again, g(a ) has previously been deﬁned n m ≤ n m m to fulﬁll g(a ) B K d(a , a ) + ǫ m ∈ f(an) · n m d g(a ), f(a ) K d(a , a ) + ǫ ⇒ m n ≤ · n m K d(a , a ) + K d(a , a ) + ǫ ≤ · n j · j m = sn + rm.

Hyperconvexity yields = . Choose g(a ) arbitrary. Fj 6 ∅ j ∈ Fj By deﬁnition, d g(aj ),fT(aj) ǫ and g is K-LipschitzT on aj j N∗ . We ≤ { } ∈ extend g to the whole of X in the same way as in the proof of Lemma 81.

We should note that we did not make use of the full strength of hyperconvexity, but restricted to countable families of balls, hence “σ-hyperconvexity” suﬃces for Lemma 83. However, as noted by Aronszajn and Panitchpakdi ([AP], Theorem 2.1), σ-hyperconvexity and separability of Z already imply full hyperconvexity. Smoothening of Lipschitz Functions 49

Another possibility to prove Lemma 83 is to show that (LipK (X, Z), d ) is ∞ hyperconvex and apply the Extension Theorem 2.3 of [AP] to the special case

id LipK,ǫ (X, Z), d LipK (X, Z), d LipK (X, Z), d . ∞ ⊇ ∞ −→ ∞ Note that the preservation of the modulus of continuity of id is of high importance to get a controlled extension. However, showing the hyperconvexity of LipK (X, Z) for separable X is again very similar to the proof of Lemma 83: Given a family f Lip (X, Z) and non-negative numbers s , k I, with I k ∈ K k ∈ an arbitrary index set, satisfying d (fk, fl) sk +sl for all k, l K, we choose ∞ ≤ ∈ a dense subset aj j J X and apply the exact same arguments to the family { } ∈ ⊆ := B K d(a , a ) : 1 n < j B (s ) : n I Fj g(an) · n j ≤ ∪ fn(aj ) n ∈ with g(a ) : inductively deﬁned. Finally extending g continuously to j ∈ Fj the whole of X returns a K-Lipschitz function which is within the intersection T of all balls Bfk (sk).

In general, the metric+ space of continuous real-valued functions C(H) of a compact Hausdorﬀ space H with metric d is not hyperconvex (mostly, it is ∞ not even metrically complete) – only in cases where H is extremally discon- nected1, hyperconvexity can be recovered. In Remark 5.1 of [AP], Aronszajn and Panitchpakdi connect the property of order completeness of the lattice structure of C(H) to hyperconvexity. In view of this argument and of Propo- sition 43, it is better comprehensible why Lipschitz function spaces are more often hyperconvex than spaces of continuous functions. The situation in the case of non-separable metric+ spaces X however is unknown to the author.

Example 84 ...... 84 We analyze an example for a simple non-hyperconvex true metric space, the Euclidean plane Z = (R2, d ). Let L> 0 and ǫ> 0 be arbitrary, X = a, b, c 2 { } with d(a, b) = d(b, c) = L, d(a, c) = 2L, and f : X R2 with → f(a) = (0, 0) , 1 3 f(b) = L + ǫ, Lǫ + ǫ2 , 2 r 4 ! and f(c) = (2L + ǫ, 0) .

We ﬁnd f Lip (X, Z). To smooth f like in the proof of Lemma 81, we ∈ 1,ǫ choose a1 = a, a2 = c, a3 = b. Hence, g(a) = f(a) = (0, 0). As 2L < g(a) g(c) 2L + ǫ, − ≤ we choose g(c) to be the point in the 2L -ball around g(a) nearest to f(c), this is g(c) = (2L, 0). Now, there is only one possibility left for g(b) to make g Lipschitz, we have to set g(b) = (L, 0). However,

d g(b), f(b) = Lǫ + ǫ2,

1i.e. the closure of each open set is open. p 50 Claim 87 which can be chosen arbitrarily large for fixed ǫ. Subsequently, the algorithm and proof fail. The proof’s algorithm of the more general Lemma 83 fails as well, in step j = 3 the intersection of the five involved balls is empty, reflecting the non- hyperconvexity of the Euclidean plane. However, in this special case (Z = R2 Euclidean, X is a true metric space of three points), the thesis of the two Lemmas still holds, one only has to choose the order of the points aj more carefully: a1 = b, a2 = a, a3 = c will yield 2L2 + Lǫ d g(a ), g(a ) = < 2L, 2 3 L + ǫ thus g Lip(X, Z), d (f, g) = ǫ, and Lip(X, Z) is ǫ-dense in Lip1,ǫ(X, Z). ∈ ∞ Example 85 ...... 85 In other examples, topological obstructions prevent the smoothening. Let X = S1 with arc length metric and of total length 2π, Z = S1 also with arc length metric, but total length 2π + 2ǫ, ǫ > 0. The identity f = id : X Z is → (1,ǫ)-Lipschitz. However, there cannot be a Lipschitz function g : X Z with → d (f, g)= O(ǫ), only O(√ǫ). ∞

Example 86 ...... 86 Take the unit sphere Z = S2 with arc length metric, and X = S1 with arc length metric and of total length 2π ǫ. Let f be the embedding of X as the equator of − Z, then f is (1,ǫ)-Lipschitz. Although there are Lipschitz-embeddings of X into Z, their distance to f is proportional to √ǫ, which inhibits a result in the sense of Lemma 81. As this argument still holds after removal of a part of the sphere near one of the poles, we conclude that there is even a topologically contractible space Z for which the thesis of Lemma 81 fails.

Example 87 ...... 87 Let G = S be a finitely generated group, which acts freely and cocompactly on h i a Riemannian manifold M. Let x M be arbitrary, then Gx is an ǫ-dense 0 ∈ 0 subset of M for some large enough δ > 0. Define f : M [0, ) by → ∞ x min dS (e, g), 7→ g G : ∈ d (gx , x) δ M 0 ≤ i.e. wordlength of (one of) the nearest elements out of Gx0. This defines a (K, ǫ)-Lipschitz function for large enough K,ǫ > 0, and hence a K-Lipschitz function f¯ : M [0, ), which represents the wordlength of G in M, at least → ∞ up to some error 2ǫ. If for example M is the Euclidean plane, G = Z2 with standard action and standard generators S, then we may choose δ = 1 and will find f to be (K = √2, ǫ = 2)-Lipschitz, and g will be a √2-Lipschitz function near to the supremum norm, depending on the chosen dense sequence (a ) R2. j ⊆ Rough ml-Isomorphisms 51

2.1.2 Algorithmic Aspects

The proof used for Lemma 81 is an online algorithm, i.e. it is possible to start the calculation of the smoothed function g without having all information about f available. Even better, to calculate the next value g(aj) the only information you need is minimal: The previously calculated values of g, the distance of aj to all previously used points, and f(aj) itself. In addition, you are free to choose the next point aj+1 freely, without risking inconsistencies. This is of importance to efficient approximations of g(a) with a = limj aj, as well as →∞ to any successive algorithm that needs Lipschitz functions as input to return a realistic result, such as deconvolution filters. Lemma 83 is superior to Lemma 81 in its conclusion, but its proof does not exhibit the same qualities. Apart from the general need for the Countable Axiom of Choice and the need to handle infinite intersections, it is also necessary to provide all information about f already to calculate g(a2). Example 84 finally provides a glimpse at the worst case scenario: Here, the choice of an admissible sequence of the points (aj) was necessary to gain a reasonable smoothening. While our example employs a finite true metric space X, it is obvious to conjecture that infinite or even uncountable true metric spaces X might require an infinite automaton to return an admissible sequence based on a given function f. The consequence is that at least some of the values of g would be non-computable numbers in the sense of [T] (in Z instead of R). In this scenario, the main interesting question, namely whether LipK (X, Z) is O(ǫ)-dense in LipK,ǫ (X, Z) or not, could even be undecidable.

2.2 Rough ml-Isomorphisms

Next we deﬁne a special version of rough isometry, suiting the lattice structure of a complete lattice with intervaluation metric+ in general, and of a Lipschitz function space in particular. The main new property will be an “approximate lattice homomorphism”. In the Lipschitz-function case, it exists in various versions, as Thomas Schick pointed out to us:

Proposition 88 ...... 88 Let X,Y be metric spaces and let κ : Lip Y Lip X be an ǫ-isometric embed- + → ding, ǫ 0 arbitrary and ﬁxed. Then the following properties are equivalent: ≥ 1. f g κf κg + ǫ for all f, g Lip Y ≤ ⇒ ≤ ∈ 2. d (κf) (κg), κ(f g) ǫ for all f, g Lip Y ∞ ∨ ∨ ≤ ∈ 3. For all f Lip Y , j J,J = some index set, holds: j ∈ ∈ 6 ∅

d κfj, κ fj ǫ and d κfj, κ fj ǫ ∞   ≤ ∞   ≤ j J j J j J j J _∈ _∈ ^∈ ^∈     52 Claim 91

Proof (3) (2): #J = 2. ⇒ (2) (1): Assume there is some x X such that (κf)(x) > (κg)(x) + ǫ. ⇒ ∈ From f g follows f g = g, thus d ((κf) (κg), κg) ǫ, in particular ≤ ∨ ∞ ∨ ≤ (κf)(x) (κg)(x) (κg)(x)+ ǫ, contradiction. ∨ ≤ (1) (3): Obviously we have f f for all k J, hence κf κ f + ⇒ k ≤ j J j ∈ k ≤ j j ǫ. We calculate the supremum over all∈ k J: κf κ f + ǫ. On the W ∈ j j ≤ j j W other hand, for every δ > 0 there is some k J with d (fk, j fj) δ. As κ is ∈ W ∞ W ≤ an ǫ-isometric embedding, this yields d (κfk, κ j fj) δ + ǫ, hence ∞ ≤ W W κ f κf + ǫ + δ κf + ǫ + δ. j ≤ k ≤ j j J j J _∈ _∈ The limit δ 0 yields the ﬁrst approximation, the other works analogously. →

We extend property (3) from Proposition 88 to allow J = , generalize to arbi- ∅ trary intervaluation metrics, and use it to deﬁne the notion of ml-isomorphisms:

Deﬁnition 89 ...... 89 Let L, L be intervaluation metric lattices. An ǫ-ml-homomorphism, ǫ 0 is ′ + ≥ an ǫ-isometric embedding κ : L L , with → ′

d κfj, κ fj ǫ and d κfj, κ fj ǫ ∞   ≤ ∞   ≤ j J j J j J j J _∈ _∈ ^∈ ^∈     for all f L, j J, J some arbitrary index set. j ∈ ∈ An ǫ-ml-isomorphism is a pair of ǫ-ml-homomorphisms κ : L L and κ : → ′ ′ L L, such that κ κ and κ κ are ǫ-near their corresponding identities. ′ → ◦ ′ ′ ◦ When we speak of an “ǫ-ml-isomorphism κ”, the corresponding κ′ shall always be implied.

Proposition 90 ...... 90 Let L be complete, and let 0 be the least element in L. For any ǫ-ml-isomorphism κ holds d(κ(0), 0) ǫ. ≤

Proof As Andreas Thom pointed out, this follows directly from Deﬁnition 89 when J = . There is also a 3ǫ-proof avoiding empty index sets: Let κ : L L ∅ → ′ be an ǫ-ml-isomorphism. We certainly know 0 κ (0) = 0 L, hence ∧ ′ ∈

d κ(0), κ(0) κκ′(0) 2ǫ. ∧ ≤ Now apply the cut law (see Deﬁnition 62, Corollary 71) to see

d κ(0) κκ′(0), κ(0) 0 ǫ ∧ ∧ ≤ and use κ(0) 0 = 0 L. ∧ ∈ Rough ml-Isomorphisms 53

Proposition 91 ...... 91 A δ-surjective ǫ-ml-homomorphism κ : Lip Y Lip X induces a (2ǫ + 2δ)-ml- → isomorphism (κ, κ′).

Proof For each f Lip X choose an element κ (f), such that d(κκ f,f) δ. ∈ ′ ′ ≤ We show that the pair (κ, κ′) defines a (2ǫ + 2δ)-ml-isomorphism. The first inequality in Definition 89 is standard in coarse geometry:

d (f, g) d (κκ′f, κκ′g) + 2δ d (κ′f, κ′g) + (ǫ + 2δ) ∞ ≤ ∞ ≤ ∞ d (f, g) d (κκ′f, κκ′g) 2δ d (κ′f, κ′g) (ǫ + 2δ) ∞ ≥ ∞ − ≥ ∞ − for all f, g Lip X. We now show that κ fulﬁlls the second and third inequality ∈ ′ as well. Both can be handled the same way:

d κ′fj, κ′ fj d κ κ′fj, κκ′ fj + ǫ ∞ ≤ ∞ ^ ^ ^ ^ d κ κ′fj, fj + ǫ + δ ≤ ∞ ^ ^ d κκ′fj, fj + 2ǫ + δ ≤ ∞ ^ ^ d κκ′fj, fj + 2ǫ + δ ≤ ∞ δ_+ 2ǫ + δ fj Lip X, j J ≤ ∀ ∈ ∈ Here we used (i) κ is ǫ-isometric embedding, (ii) κκ′ is near identity, (iii) κ is ml-homomorphism, (iv) Proposition 72, (v) κκ′ is near identity.

Finally we show that κ′κ is (ǫ + δ)-near identity:

d (κ′κf,f) d (κκ′(κf), (κf)) + ǫ δ + ǫ f Lip X. ∞ ≤ ∞ ≤ ∀ ∈

Lemma 92 ...... 92 Let L and L′ be intervaluation metric+ lattices with least elements, and let the family of all bounded completely ml-irreducible elements bcmli(L′) be a base of L (see Deﬁnitions 73 and 76). Then any ǫ-ml-isomorphism κ : L L , ǫ 0 ′ → ′ ≥ maps bcmli(L) 6ǫ-near bcmli(L′).

Proof Let p bcmli(L) be some bounded completely ml-irreducible element. ∈ Represent κ(p) via elements q bcmli(L ), j J, J some non-empty index j ∈ ′ ∈ set. Let δ > 0 be arbitrary. Then we have

d κ(p), qj = 0 apply κ′ ∞ | _ d p, κ′(qj) 3ǫ. ⇒ ∞ ≤ As p is completely ml-irreducible,_ we know that there exists k J such that ∈ d (p, κ′(qk)) 3ǫ + δ apply κ ∞ ≤ | d (κ(p), qk) 5ǫ + δ. ⇒ ∞ ≤ 54 Claim 94

Case 1: ǫ> 0. Choose δ = ǫ. Case 2: ǫ = 0. The preceding argument yields a sequence of bounded completely ml-irreducible elements qk metrically converging to κ(p). As of Proposition 75, κ(p) must be bounded and completely ml-irreducible as well.

2.3 Λ-Functions

Lip X is no algebra, like e.g. C(X). Thus we cannot give a linear base of functions and reconstruct Lip X by linear combinations. However, we can use the lattice structure to give a base in the sense of Deﬁnition 76 for Lip X: Minimal Lipschitz functions with a given value at a single point.

Definition 93 ...... 93 Let x, y X and r [0, ] be arbitrary. Define Λ(x, r) Lip X by ∈ ∈ ∞ ∈ Λ(x, r)(y) := r d(x, y) 0. − ∨ Note that this definition applies to r = or d(x,y)= as well: If d(x,y)= , ∞ ∞ ∞ we have Λ(x, r)(y) = 0, and if r = : ∞ : d(x,y) = Λ(x, )(y) = ∞ 6 ∞ ∞ 0 : d(x,y)= ∞ Λ-functions with r = will be called infinite, else bounded. Infinite Λ-functions ∞ are infinitely high characteristic functions for X’s components.

Proposition 94 ...... 94 Consider x, y X and r, s [0, ]. Then holds: ∈ ∈ ∞ r s : d(x,y) r s ∨ ≥ ∧ d (Λ(x, r), Λ(y,s)) = r s + d(x,y) : d(x,y) r s < ∞  | − | ≤ ∧ ∞ 0 : d(x,y) < r s =  ∧ ∞ r s + d(x,y) ≤ | − |

Proof Note that if d(x,y) = r s the ﬁrst and second case coincide, as ∧ r s = (r s) (r s). Assume without restriction r s. Consider | − | ∨ − ∧ ≤ fz := 0 (r d(x, z)) 0 (s d(y, z)) z X ∨ − − ∨ − ∀ ∈ d := d (Λ(x, r), Λ(y,s)) = f(z). ∞ z X _∈ Let us start with the inﬁnite cases. If r = s = d(x,y) = , we get d = ∞ ∞ on both sides. If r = s = , d(x,y) = , we get d = 0. This is correct, as ∞ 6 ∞ in this case the Λ-functions are equal. If s = , r = we get d = again, ∞ 6 ∞ ∞ Λ-Functions 55

Figure 2.1: The d -distance between two Λ-functions is determined by their ∞ difference evaluated at the maximum point of the greater function, see Prop. 94. for each variant of d(x,y). If r, s = but d(x,y) = , the two Λ-functions 6 ∞ ∞ have different components as support, and thus d becomes the maximum of the differences, this is s. Now we assume r, s, d(x,y) = . First case: r d(x,y). Then we have 6 ∞ ≤ d fy = s 0 (r d(x, y)) = s. ≥ − ∨ − In addition, we have Λ(x, r)(z) [0, r] and Λ(y,s)(z) [0 ,s], thus fz r s = s, ∈ ∈ ≤ ∨ hence d = s. Second case: d(x,y) r. We find ≤ d fy = s 0 (r d(x,y)) = s r + d(x,y), ≥ − ∨ − − and finally: fz = r d(x, z) s + d(y, z) r s + d(y, z) d(x, z) | − − | ≤ | − | | − | r s + d(x,y) = s r + d(x,y) z X ≤ | − | − ∀ ∈

Corollary 95 ...... 95 For all x,y X holds: ∈ d(x,y) = lim d Λ(x, r), Λ(y, r) r ,r= ∞ →∞ 6 ∞ Proof Follows directly from Proposition 94.

This Corollary points us at an interesting aspect of Λ-functions: When we analyze the metric+ space Xr := Λ(x, r) : x X with metric d for a fixed { ∈ } ∞ r R , we find it to be naturally isometric to (X, d ) with the cut-off-metric ∈ >0 r dr(x,y) := r d(x,y) for all x,y X. Only in the limit r , d will restore ∧ ∈ →∞ ∞ the full metric of X. Ironically, d obviously cuts away the coarse, large- ∞ scale information of X (in which we are primarily interested) and preserves the topological, small-scale information. The large-scale information of X is still present, but more subtle to access. 56 Claim 97

Proposition 96 ...... 96 For all f Lip X holds: f = Λ(x,f(x)), where the latter is a pointwise ∈ x X maximum, not only supremum (cp.∈ Figure 2.2). If X is complete, the set W A := Λ(x,f(x)) : x X 0 is (topologically) closed. { ∈ } ∪ { }

Proof Deﬁne g := Λ(x,fx). Clearly, we have z X : fz gz, as x X ∀ ∈ ≤ fz = Λ(z,fz)(z). We now∈ observe that W fz Λ(x,fx)(z) x, z X. ≥ ∀ ∈ For d(x, z) fx, this is clear. For d(x, z) fx, this follows from Lipschitz ≥ ≤ continuity (fz fx d(x, z)). ≥ − Furthermore, we notice that we deal with pointwise maxima: Each supremum of Λ(x,fx)(z) x X is taken by Λ(z,fz)(z)= fz. { } ∈ Let Λ(x ,fx ) be any sequence converging to g Lip X, x X, j N . First j j ∈ j ∈ ∈ ∗ we notice

fxj = d (0, Λ(xj, fxj)) d (0, g) d (g, Λ(xj, fxj)) ∞ ≥ ∞ − ∞ and fxj d (0, g) + d (g, Λ(xj, fxj)), ≤ ∞ ∞ hence fxj d (0, g). Assume g = 0 and ﬁnite. Then there is x X with → ∞ 6 ∈ gx > 0 and fxj must have a lower bound R> 0 for large enough j. By Cauchy criterion there is N N such that for all j, k > N we have ∈ ∗ 1 d (Λ(xj ,fxj), Λ(xk,fxk)) R < fxj fxk. ∞ ≤ 2 ∧ Due to Proposition 94 we conclude that for large enough j, k

d (Λ(xj, fxj), Λ(xk, fxk)) = fxj fxk + d(xj , xk) 0. ∞ | − | → Thus fx as well as x are Cauchy-sequences. As X and [0, ] are metrically j j ∞ complete, we ﬁnd x′ := lim xj. As f is continuous, we have fx′ = lim fxj, and Λ(x ,fx ) A. Now we only have to show g = Λ(x ,fx ). But this is clear, as ′ ′ ∈ ′ ′ for large enough j we have

d (Λ(x′, fx′), Λ(xj, fxj)) fx′ fxj + d(x′, xj) 0 + 0. ∞ ≤ | − | → Now assume g to be infinite (i.e. x : gx = ). Then fx has to be infinite ∃ ∞ j as well for large enough j (there is no non-trivial convergence to infinity in the chosen metric on [0, ) and Proposition 94 shows d(x ,x ) < for large ∞ j k ∞ enough j, k). Hence Λ(xj, fxj) = Λ(xk, fxk) = g.

Corollary 97 ...... 97 The family of all bounded Λ-functions is a metrically closed base in the sense of Deﬁnition 76. Λ-Functions 57

Figure 2.2: Proposition 96 demonstrates how to decompose arbitrary 1- Lipschitz functions into a supremum of Λ-functions.

Proof We note that Λ(x, ) is the supremum of all Λ(x, r) with r [0, ), ∞ ∈ ∞ i.e. we do not need inﬁnite Λ-functions to represent functions from Lip X. In addition, the family of all bounded Λ-functions is an intersection of the closed set of all Λ-functions and the closed component of the zero-function, thus it is closed as well.

We make some more use of the black magic of Proposition 72:

Proposition 98 ...... 98 For all ǫ-isometries η : X Y and f Lip Y holds: → ∈

d Λ(x,fηx), Λ(η′y,fy) ǫ ∞   ≤ x X y Y _∈ _∈  

Proof We observe that d := d x X Λ(x,fηx), y Y Λ(η′y,fy) can be ∞ rewritten to ∈ ∈ W W

d = d Λ(x,fηx), Λ(η′y,fy) ∞   (x,y) J (x,y) J _∈ _∈   where J := (x,y) X Y : y = ηx or x = η y : Each element of X (respec- { ∈ × ′ } tively Y ) appears at least once in J, and multiple instances do not matter, as is idempotent. Now Proposition 72 yields:

W d d Λ(x,fηx), Λ(η′y,fy) ≤ ∞ (x,y) J _∈ Let (x,y) J. Case 1: y = ηx. Then ∈ d Λ(x,fηx), Λ(η′y,fy) = d Λ(x,fηx), Λ(η′ηx,fηx) ∞ ∞ d(x, η′ηx) ǫ ≤ ≤ Case 2: x = η′y:

d Λ(x,fηx), Λ(η′y,fy) = d Λ(η′y,fηη′y), Λ(η′y,fy) ∞ ∞ fηη′y fy ǫ ≤ | − | ≤ 58 Claim 100

Figure 2.3: “Lipschitzization”g ¯ of a roughly Lipschitz function g (see Lemma 99). Three of the composing Λ-functions are shown.

We make a ﬁrst use of the notions we derived. We will ﬁrst show a smoothening theorem in the manner of Section 2.1, but using Λ-functions as new tools. We will then lift each ǫ-isometry η : X Y to an ǫ-isometryη ¯ : Lip Y Lip X. → → Even better,η ¯ is an ǫ-ml-isomorphism, and is near f f η. 7→ ◦

Lemma 99 ...... 99 Let f Lip (X, [0, ]). Deﬁne ∈ 1,ǫ ∞ f¯ := Λ(x,fx). x X _∈ Then f and f¯ are ǫ-near (cp. Figure 2.3).

Proof We observe that f(y) is never larger than x X Λ(x,fx)(y) for all y X. So we have ∈ ∈ W

d f, Λ(x,fx) = Λ(x,fx)(y) f(y) ∞ − x X ! x,y X _∈ _∈ and furthermore f(y) : d(x,y) f(x) Λ(x,fx)(y) f(y) = − ≥ . − f(x) f(y) d(x,y) : d(x,y) f(x) − − ≤ As f(x) f(y) d(x,y) ǫ and f(y) 0 ǫ we conclude the statement. − − ≤ − ≤ ≤ (Note that each negative value is surpassed by at least one non-negative value, i.e. f(y) never occurs after taking the supremum.) −

Proposition 100 ...... 100 If η : X Y is an ǫ-isometry, and κ : Lip Y Lip X any mapping which is → → δ-near f f η, then κ is a (2ǫ + 2δ)-ml-homomorphism. 7→ ◦

Proof (i) We show d (f, g) d (κf, κg) 2ǫ + 2δ for all f, g Lip X. | ∞ − ∞ | ≤ ∈ We have

d (κf, κg) d (f η, g η) 2δ. ∞ − ∞ ◦ ◦ ≤

Λ-Functions are Completely ml-Irreducible 59

As next we notice d (f η, g η) d (f, g). Now let y Y be arbitrary, ∞ ◦ ◦ ≤ ∞ ∈ x := η y X. Then fηx fy ǫ as f is 1-Lipschitz. Hence ′ ∈ | − |≤ fy gy fηx gηx + 2ǫ d (f η, g η) + 2ǫ | − | ≤ | − | ≤ ∞ ◦ ◦ d (f, g) d (f η, g η) + 2ǫ. ⇒ ∞ ≤ ∞ ◦ ◦ (ii) For J = we observe that d (κ(0), 0 η) δ and 0 η = 0, as well as ∅ ∞ ◦ ≤ ◦ d (κ( ), η) δ and η = . Hence, assume J = . We know ∞ ∞ ∞ ◦ ≤ ∞ ◦ ∞ 6 ∅

d κ fj , (fj η) = d κ fj , fj η δ ∞ ◦ ∞ ◦ ≤ ^ ^ ^ ^ as the inﬁmum is calculated pointwise. Hence, with Proposition 72:

d (κfj), κ fj d (κfj), (fj η) + δ ∞ ≤ ∞ ◦ ^ ^ ^ ^ d (κfj,fj η)+ δ ≤ ∞ ◦ _ǫ + δ. ≤ Same for supremum.

Theorem 101 (= Th. 2) 101 Given an ǫ-isometry η : X Y , η¯(f) := f η deﬁnes a 4ǫ-ml-isomorphism → ◦ from Lip Y to Lip X.

Proof Let f Lip Y be arbitrary. f η satisﬁes ∈ ◦ d(fηx, fηy) d(ηx, ηy) d(x, y) + ǫ. ≤ ≤ Hence, f η andη ¯(f) are ǫ-near (Lemma 99). However,η ¯(f) is in Lip X, as it ◦ is a supremum of Lipschitz functions. Thus we can apply Proposition 100 to η¯ : Lip Y Lip X. Same holds for η (Deﬁnition 31). It remains to show that → ′ η¯ η and η η¯ are near their respective identities. ◦ ′ ′ ◦ We already saw thatη ¯(η (f)) is ǫ-near (η f) η. Similarly η f is ǫ-near f η ′ ′ ◦ ′ ◦ ′ and thus (η f) η is ǫ-near f η η. Finally, η η is ǫ-near identity, and as f ′ ◦ ◦ ′ ◦ ′ ◦ is 1-Lipschitz, f η η is ǫ-near f, too. All this adds up to 3ǫ. Same for η η¯. ◦ ′ ◦ ′ ◦

2.4 Λ-Functions are Completely ml-Irreducible

Lemma 102 ...... 102 Consider p Lip Y with Y complete. Then the following are equivalent: ∈ 1. p is a bounded Λ-function, i.e. x Y, r [0, ) : p = Λ(x, r), ∃ ∈ ∈ ∞ 2. p is a bounded completely ml-irreducible element (see Deﬁnition 73). 60 Claim 102

Proof For convenience, we repeat the deﬁnition of a completely ml-irreducible element p: For all (fj)j J Lip Y with an arbitrary, non-empty index set J, ∈ ⊆ and any R [0, ] holds: ∈ ∞

d p, fj R δ > 0 j J : d (p,fj) R + δ ∞   ≤ ⇒ ∀ ∃ ∈ ∞ ≤ j J _∈   The case R = is trivial. Hence, assume R to be ﬁnite. ∞ (1) (2): Let (fj)j J Lip Y and R 0 be such that d (p, j J fj) R ⇒ ∈ ⊆ ≥ ∞ ≤ holds. Choose δ > 0 arbitrary and p = Λ(y,s), for some y Y , s [0∈, ). As ∈ W∈ ∞ d p(y), f (y) R p(y) R δ < f (y), j ≤ ⇒ − − j _ _ there has to be a k J such that p(y) R δ

f (x) f (y) d(x,y) > p(y) Wd(x,y) R δ. k ≥ k − − − − Case 1: p(y) d(x,y). Then we have p(x)= p(y) d(x,y), and ≥ − f (x) > p(x) R δ. k − − Case 2: p(y) d(x,y). Then p(x)=0 and ≤ f (x) 0 > p(x) R δ k ≥ − − holds trivially. On the other hand, we have

f (x) f (x) p(x)+ R< p(x)+ R + δ x Y k ≤ j ≤ ∀ ∈ j J _∈

and thus d (fk,p) < R + δ. The proof is illustrated by Figure 2.4. ∞ (2) (1): The family of all bounded Λ-functions is a metrically closed base ⇒ according to Corollary 97, and due to Proposition 78, the bounded completely ml-irreducible elements form a subset in them.

The formula

d p, fj R δ > 0 j J : d (p,fj) R + δ ∞   ≤ ⇒ ∀ ∃ ∈ ∞ ≤ j J _∈   of Deﬁnition 73 does not hold for δ = 0, as one can see in Lemma 102: As a

counter-example, insert Λ(y, 1) = r (0,1) Λ(y, r). This is exactly the diﬀerence between a completely irreducible and∈ a completely ml-irreducible element. W Recalling the short note after Corollary 95, the metric information of Y is encoded in the Λ-functions and the distances between them. However, these Inducing Rough Isometries 61

Figure 2.4: When approximating a Λ-function p by Lipschitz functions fj, one of the functions (here f1) must approximate the maximum point of p. This function may not decrease too fast (Lipschitz!), and may not increase too fast, as it is bounded from above by the approximation of p, hence it already approximates p on its own, see Lemma 102. functions are at ﬁrst sight just some arbitrary subset of Lip Y and thus there seems to be no hope for the metric+ space (Lip Y, d ) to hold the full infor- ∞ mation about Y ’s metric. The preceding Lemma now explains to us that the (bounded) Λ-functions are not arbitrary at all – they have a speciﬁc, lattice theoretic property that distinguishes them from the remaining functions. Hence, in some sense the metric information of Y is now part of the combined metric and lattice structure of Lip Y .

Corollary 103 ...... 103 Let X, Y be complete, ǫ 0 and κ : Lip Y Lip X an ǫ-ml-isomorphism. ≥ → Then κ maps bounded Λ-functions 6ǫ-near bounded Λ-functions.

Proof Follows from Lemmas 92 and 102.

The preceding Corollary is the critical point in our analysis: We can use Λ- functions as building blocks for Lipschitz functions, as Proposition 96 tells us. From Corollary 103 we now know that these building blocks (or, at least, the bounded versions) behave sensible under ǫ-ml-isomorphisms κ, such that we only have to understand how they are mapped by κ to reconstruct all other Lipschitz functions. In particular, as they are strongly connected to the underlying spaces, they allow us to deﬁne mappings between them, as will be shown next.

2.5 Inducing Rough Isometries

In this section, we show the reversal of Theorem 101: Given an ǫ-ml-isomorphism κ we construct a rough isometry η such thatη ¯ is near κ. 62 Claim 104

Lemma 104 ...... 104 Let X, Y be complete, and ǫ 0 arbitrary. Let κ : Lip Y Lip X be an ≥ → ǫ-ml-isomorphism, ǫ 0. Then there is a map η : X Y such that ≥ →

d Λ(ηx,r), κ′(Λ(x, r)) 59ǫ ∞ ≤ for all x X, r [0, ]. For r [38ǫ, ), we may replace “59ǫ” by “43ǫ”. ∈ ∈ ∞ ∈ ∞

Proof In the following proof, the first two cases will deal with ǫ> 0 and finite r, the third with ǫ = 0 and finite r and the fourth with r = . ∞ Case 1 and 2: For each x X, choose η(x) Y and s [0, ) such that ∈ ∈ x ∈ ∞ Λ(ηx,sx) is 6ǫ-near κ′Λ(x, 22ǫ) (use Corollary 103). Case 1: ǫ > 0, r [38ǫ, ). Let Λ(x , r ) be 6ǫ-near κ Λ(x, r). Then by ∈ ∞ ′ ′ ′ Proposition 90 holds

d (0, Λ(x, r)) = r d (0, κ′Λ(x, r)) r 2ǫ ∞ ⇒ ∞ − ≤ r′ r 8ǫ. ⇒ | − | ≤ In the same way, we have

d (0, Λ(ηx,sx)) d (0, κ′Λ(x, 22ǫ)) 6ǫ ∞ − ∞ ≤ sx 22ǫ 8ǫ. ⇒ − ≤

We now take a look at d (Λ(x, r), Λ(x, 22ǫ)) = r 22ǫ (as r 22ǫ) ∞ − ≥ d (κ′Λ(x, r), κ′Λ(x, 22ǫ)) (r 22ǫ) ǫ ⇒ ∞ − − ≤ d (Λ(x′, r′), Λ(ηx,sx)) (r 22ǫ) 13ǫ. ⇒ ∞ − − ≤

Now we calculate d := d (Λ(x′, r′), Λ(ηx,sx)) by hand. From Proposition 94, ∞ d could be r s or d(x ,ηx)+ r s . We know ′ ∨ x ′ | ′ − x|

s 8ǫ + 22ǫ = 30ǫ r 8ǫ r′, x ≤ ≤ − ≤ hence r s = r . However, as d r 22ǫ + 13ǫ = r 9ǫ, but r r 8ǫ, d ′ ∨ x ′ ≤ − − ′ ≥ − cannot be r′ (here we use ǫ> 0). Remains

d = d(x′,ηx)+ r′ s with d (r 22ǫ) 13ǫ. | − x| | − − | ≤ As shown above, r s , hence ′ ≥ x

d(x′, ηx) r 22ǫ + 13ǫ r′ s = r 22ǫ + 13ǫ r′ + s ≤ − − | − x| − − x r 22ǫ + 13ǫ r + 8ǫ + 22ǫ + 8ǫ = 29ǫ. ≤ − − This, and r r 8ǫ, yield | ′ − |≤

d (Λ(ηx,r), Λ(x′, r′)) d(x′,ηx)+ r r′ 37ǫ ∞ ≤ | − | ≤ d (Λ(ηx,r), κ′Λ(x, r)) 43ǫ. ⇒ ∞ ≤ Inducing Rough Isometries 63

Figure 2.5: The function Λ(x′, r′) in the proof of Lemma 104 is already determined up to nearness by its distance to two other functions: the zero function and Λ(ηx,sx). This shows: A Λ-function Λ(y,s) is not only mapped near another Λ-function Λ(y′,s′), but y′ only depends on y and s′ only depends on s (modulo some multiples of ǫ).

Case 2: ǫ> 0, r [0, 38ǫ). Obviously, ∈

d (Λ(ηx,r), κ′Λ(x, r)) d (Λ(ηx,r), Λ(ηx,sx)) ∞ ≤ ∞ + d (Λ(ηx,sx), κ′Λ(x, 22ǫ)) ∞ + d (κ′Λ(x, 22ǫ), κ′Λ(x, r)) ∞ r s + 6ǫ + ǫ + r 22ǫ ≤ | − x| | − | As r [0, 38ǫ) and s [14ǫ, 30ǫ] (see above), we receive r s 30ǫ and ∈ x ∈ | − x| ≤ r 22ǫ 22ǫ. This adds up to 59ǫ. | − |≤ Case 3: ǫ = 0, r [0, ). As of Corollary 103, for all x X we can choose ∈ ∞ ∈ η(x) such that κ Λ(x, 1) = Λ(ηx,s ) for some s [0, ). From Proposition 90 ′ x x ∈ ∞ we see κ (0) = 0, hence s = 1. Now, let r [0, ) be arbitrary. Let x Y , ′ x ∈ ∞ ′ ∈ r [0, ] such that κ Λ(x, r) = Λ(x , r ). Clearly, from the distance to 0 we ′ ∈ ∞ ′ ′ ′ again have r′ = r. From

d (Λ(x, 1), Λ(x, r)) = r 1 ∞ | − | 64 Claim 106 we conclude

d (Λ(ηx, 1), Λ(x′, r)) = r 1 . ∞ | − | Due to Proposition 94 this can happen iﬀ (a) r 1 = 1 r or (b) r 1 = r 1 | − | ≥ | − | ≥ or (c) d(x′,ηx) = 0. Case (c) proves our statement, case (b) cannot happen: r 1 = r iﬀ r = 1 , which contradicts r 1. So, assume case (a). Then | − | 2 ≥ r = 2, which contradicts r 1, or r = 0. But the case r = 0 is trivial, as we ≤ already saw from Proposition 90 that

κ′Λ(x, 0) = 0 = Λ(ηx, 0).

Case 4: r = . We know Λ(x, ) = Λ(x,s). Using our result for ∞ ∞ s [38ǫ, ) ﬁnite r, we conclude ∈ ∞ W

d κ′ Λ(x,s), Λ(ηx,s) 1ǫ + 43ǫ. ∞   ≤ s [38ǫ, ) s [38ǫ, ) ∈ _ ∞ ∈ _ ∞   Apply Λ(ηx,s) = Λ(ηx, ) to see that κ Λ(x, ) is indeed 44ǫ-near s [38ǫ, ) ∞ ′ ∞ Λ(ηx, ).∈ ∞ ∞W

Lemma 105 ...... 105 η : X Y as deﬁned in the proof of Lemma 104 is an 88ǫ-isometry. → Proof From Corollary 95 follows d(ηx,ηy) = lim d Λ(ηx,r), Λ(ηy,r) . r ,r= ∞ →∞ 6 ∞ Applying Lemma 104 for large enough r yields:

d Λ(ηx,r), Λ(ηy,r) d κ′Λ(x, r), κ′Λ(y, r) 2 43ǫ ∞ − ∞ ≤ · and of course

d κ′Λ(x, r), κ′Λ(y, r) d Λ(x, r), Λ(y, r) ǫ. ∞ − ∞ ≤ Hence

d(ηx,ηy) d(x,y) 87ǫ, − ≤ i.e. η is a rough isometric embedding. Just as η was constructed from κ , we ′ construct η from κ. It remains to show that η η and η η are near identities. ′ ◦ ′ ′ ◦ Again, we make use of Corollary 95:

d(ηη′x, x) lim d Λ(ηη′x, r), Λ(x, r) = 0 − r ,r= ∞ →∞ 6 ∞ d(ηη′x, x) lim d κ′κΛ(x, r), Λ(x, r) 2 43ǫ + ǫ ⇒ − ∞ ≤ · d(ηη′x,x) 88ǫ ⇒ ≤ Same for η η. ′ ◦ Other Metrics for Lip X 65

Theorem 106 (= Th. 3) 106 Let X,Y be complete metric spaces and ǫ 0. For each ǫ-ml-isomorphism + ≥ κ : Lip(Y ) Lip(X) there is an 88ǫ-isometry η : X Y , such that κ is → → 61ǫ-near η¯ : f f η. 7→ ◦

Proof Construct η as in Lemma 104. It is an 88ǫ-isometry due to Lemma 105. It remains to show that κ is nearη ¯: Let f Lip Y be arbitrary. Represent f ∈ via Λ-functions as in Proposition 96. Obviously,

d κ Λ(y,fy), Λ(η′y,fy) 1ǫ + 59ǫ ∞   ≤ y Y y Y _∈ _∈   due to Lemma 104. Apply Proposition 98.

2.6 Other Metrics for Lip X

We have seen in Chapter 1 that a lattice like Lip X can be equipped with a large variety of diﬀerent metrics. We want to take a look at some of them.

Example 107 (Discrete Metric or Topology) ...... 107 The discrete metric is an ultravaluation of Example 59 for a positive and constant function κ : X [0, ]. We have further seen that complete irreducibil- → ∞ ity and complete ml-irreducibility are the same in this metric in Section 1.4. More generally, when the topology derived from the metric is discrete, complete ml-irreducibility implies complete irreducibility. However, as long as X is non- empty, the only completely irreducible element in Lip X is the zero function.

This is reﬂected in the inability to state a Smoothening Theorem for (Lip X, ddis) in the way of Theorem 82: Lip X and Lip Y are trivially always ǫ-ml-isometric for ǫ > diam Lip X diam Lip Y ; otherwise, they must be perfectly lattice iso- ∨ morphic. For the most basic example, #X = 1 and #Y = 2, this already fails.

Example 108 (L1-Metric) ...... 108 The most simple valuation metric on Lip X for a measure space (X, µ) is the L1-metric from Example 56, and a very canonical choice for many applications in Lip X; however, it is not a good choice in handling with coarse geometry. Let X = Y = R2 be the Euclidean plane, ǫ > 0 arbitrary, and

η : X Y → (r, φ) (0 (r ǫ), φ) 7→ ∨ − in polar coordinates. η is an “additive contraction” of the Euclidean plane, where each point moves a distance ǫ nearer to the origin, while all elements inside the ǫ-ball around the origin are directly mapped to it. η obviously is 66 Claim 110 a 2ǫ-isometry. Let µ be the standard Lebesgue measure on X and Y . Let f = Λ (0, 0), r for r [0, ], then d (0, f) is exactly the volume of a cone of ∈ ∞ 1 height r and radius r, i.e. r3 π/3. On the other hand, f η already is Lipschitz · ◦ (hence it equals f η), and is the frustrum of a cone of height r, radius r + ǫ, ◦ and projected height r + ǫ, hence its distance to zero is: π d (0, f η) = (r + ǫ)3 ǫ3 1 ◦ 3 · − which is far, far away from d1(0, f). This problem cannot be avoided even when switching the codomain to an interval of R, and each other Lp-metric for 1 ǫ and f = Λ(j, r ǫ) with j J = j − ∈ B (ǫ). Then d(p, f ) is ǫ3 π/3, but d(p, f ) = (r3 (r ǫ)3) π/3, hence (0,0) j · j − − · of order r2. On the other hand, we know from Proposition 78 and Corollary W 97 that all completely ml-irreducible elements must be in the family of bounded Λ-functions (this property is independent of the metric). Hence, in this special case, there is no completely ml-irreducible element other than the zero function.

Example 109 (Lipschitz Norm) ...... 109 Let us consider Weaver’s Lipschitz norm from Example 69. In addition to the supremum metric, it additionally measures the Lipschitz constant of the diﬀerence function. In Lip X, this is bounded by 1, and should not provide a problem for ǫ > 1. Hence, choose ǫ < 1, furthermore #X = 1, Y = a, b { } with diameter ǫ, and η : X Y simply pt a. Take f : Y [0, ] to be → 7→ → ∞ (0, ǫ). Then the Lipschitz norm of f clearly is 1, but f η = 0. Thus there || ◦ ||L cannot be an analog of Theorem 82 for the Lipschitz norm for ǫ < 1 (or < K, respectively). A similar theme would be to explore the rough isometries of Haj lasz-Sobolev spaces ([He], chapter 5). These are subsets of Lp function spaces, with a norm similar to the Sobolev norm and hence similar to Weaver’s Lipschitz norm. Sim- ilar counter-examples as the one above should hold for Haj lasz-Sobolev spaces.

Example 110 (Peak Metric) ...... 110 We take a look at the peak metric defined in 61. This ultrametric is very similar to the discrete metric, in that no non-trivial Λ-function is completely ml- irreducible (take f = Λ(x, r ) with r r to approach p = Λ(x, r), with j j j ր d(p, fj) = r). As counter-example we make use of the spaces in Example 109, but with f = (r, r + ǫ) and g = (r, r) for some large enough r [0, ). Then ∈ ∞ d(f, g) = r + ǫ, but d(f η, g η) = 0. ◦ ◦ Example 111 (Basepoint Metrics) ...... 111 We finish with the inner and outer basepoint metrics we defined in 61, based at x X. The outer basepoint metric shares certain similarities with the dis- 0 ∈ crete metrics, but there are lots of accumulation points in Lip X (every function f which takes value 0 at least once), and the metric is unbounded. Choose Quasi-Isometries 67

R (0, ) large enough, take X = [0, R] R, Y = [0, R + ǫ], η the ∈ ∞ ⊆ canonical embedding (which is an ǫ-isometry). Choose f = Λ(R + ǫ, ǫ). Then d(f, 0) = R + ǫ, but d(f η, 0 η) = 0. ◦ ◦ For the inner basepoint metric, the inverse of the above counter-example applies: Consider Y = [0, ǫ] with basepoint 0, and X = 0 , with standard embedding { } η. Take f = Λ(ǫ, ǫ). Then d(f, 0) = 1 (as f and 0 diﬀer at each point x > 0), but d(f η, 0 η) = 0. ◦ ◦ In both cases there again are no non-trivial completely ml-irreducible elements.

In view of these Examples, we now feel conﬁdent to state the following conjecture:

Conjecture 112 ...... 112 The statement “X is ǫ-isometric to Y , then (Lip X, dX ) and (Lip Y, dY ) are 6 ǫ-ml-isomorphic” is true if and only if each Λ-function in Lip X and Lip Y is completely ml-irreducible relative to dX and dY , respectively. (There might be trivial exceptions.)

The use of other types of functions mostly is of similar failure:

Example 113 ...... 113 Take X = 0 Y = 0, 1 R and η the inclusion, ǫ = 1. The me- { } ⊆ { } ⊆ tric spaces of [0, ]-valued continuous functions C(X) and C(Y ) with sup- + ∞ norm are isomorphic to [0, ] and [0, ]2 respectively, which are not roughly ∞ ∞ isometric.

2.7 Quasi-Isometries

An obvious generalization is the question, whether similar statements as those of Theorems 101 and 106 hold for quasi-isometries instead of rough isometries. Although many ideas still work in the context of quasi-isometries, a function’s Lipschitz constant is distorted in the process of Lemma 99. Hence there happens to be a “mixing” of the Lipschitz function spaces LipK X, which creates deep problems. Indeed, it is always possible to split a quasi-isometry η : X Y into → two rough isometries η : X X and η : Y Y and a bilipschitz mapping X → 0 Y 0 → η : X Y by introducing suﬃciently ǫ-dense and ǫ-discrete nets X X ′ 0 → 0 0 ⊆ and Y Y , so the problem completely reduces to the problem for bilipschitz 0 ⊆ mappings.

Example 114 ...... 114 Let X = R, and η : X X be the function which is the identity on the nega- → tives and doubles each non-negative argument. η is bilipschitz with bilipschitz- constant 2. Let f be any 1-Lipschitz function, then f η can be anything between ◦ 1-Lipschitz and 2-Lipschitz. To allow for further comparisons between diﬀerent 68 Claim 116

functions, one has to change the function spaces: Just switching to Lip2 X in this case will not suﬃce, as η(Lip1 X) is not roughly dense in Lip2 X. And combining all of them into K 0 LipK X is not compatible with the Lipschitz- ization of Lemma 99, as the necessary≥ ǫ cannot be bounded. However, there are S chances to deﬁne generalized Lipschitz functions of the kind

d(fx, fy) c(x, y) d(x, y) ≤ · 1 where c : X X (c− , c ) is a ﬁxed function, bounded by c [1, ). c may × → 0 0 0 ∈ ∞ absorb the distortion by the quasi-isometry, but this approach uglily depends on the quasi-isometry itself and is of inferior expressiveness.

2.8 Scaling limits

Deﬁnition 115 ...... 115 Let the rough distance dR(X, Y ) between two metric+ spaces X and Y be the inﬁmum over all ǫ 0 such that X and Y are ǫ-isometric, or if there are ≥ ∞ none. If dR(X, Y ) = 0, the spaces X and X′ will be called a pseudo-isometric.

The rough distance fulfills a triangle-inequality, as concatenation of an ǫ- and a δ-isometry is an (ǫ + δ)-isometry. It is closely related to the Gromov-Hausdorff- 1 Distance for compact spaces, but may differ in a variable between 2 and 2 (i.e. they are Lipschitz-equivalent, see e.g. [Gv2], Proposition 3.5). Pseudo-isometry is a little bit less than isometry. However, they are equivalent if only compact spaces are compared (e.g. [P], [Gv2]), or if we deal with simple graphs, due to their integer metric. A nice article about scaling limits, Gromov-Hausdorff distances and quasi-isometries in the case of graphs and Cayley graphs is [Re].

Deﬁnition 116 ...... 116 Let Met∗ be the (non-small) groupoid of all pseudo-isometry-classes of metric+ spaces with ǫ-isometries as morphisms. dR is a metric+ on Met∗ in a natural way (apart from the fact that Met∗ is no set).

Each of the components of Met∗ can be endowed with a metric and topology, with the only drawback of being proper classes. This “topology” allows us to deﬁne the convergence of metric+ spaces to another metric+ space, up to pseudo-isometry. Met∗ is complete in this “topology” (cf. [P], Proposition 6, the proof works in non-compact and non-separable cases as well).

Deﬁnition 117 ...... 117 For any ℓ > 0 deﬁne s : Met Met by ℓ ∗ → ∗ s [(X, d)] := [(X, ℓ d)], ℓ · Scaling limits 69 which scales each metric space in Met by the factor ℓ (with ℓ := ). + ∗ ·∞ ∞ This operation clearly is compatible with pseudo-isometry. Let [X] be a class of spaces in Met∗. If the limit

s [X] := lim sℓ [X] ℓ 0 → exists for all sequences ℓ 0, then s [X] (resp. all members of s [X]) is called → the (strong) scaling limit of [X].

We now want to apply Theorem 101.

Corollary 118 ...... 118 Let X, Y be some metric+ spaces, such that Y is a strong scaling limit of X (Y is unique up to pseudo-isometry). Then there is a strong scaling limit of Lip X, and it is pseudo-isometric to Lip Y . (“The scaling limit of the Lipschitz space is the Lipschitz space of the scaling limit, modulo pseudo-isometry.”)

Proof As d (Y, s X) 0 for ℓ 0, there are ǫ -isometries η : s X Y R ℓ → → ℓ ℓ ℓ → with ǫ 0. These induce 4ǫ -ml-isomorphismsη ¯ : Lip Y Lip s X, which ℓ → ℓ ℓ → ℓ are in particular 4ǫ -isometries. Hence, d (Lip Y, Lip s X) 0. Proper ℓ R ℓ → rescaling of the associated Lipschitz functions further shows sℓ Lip X is naturally isometric to Lip sℓ X, hence sℓ Lip X Lip Y up to pseudo-isometry. →

Note that we can restrict to a set of Met∗ when calculating a scaling limit. Thus, we can make use of Banach’s Fixed Point Theorem.

We may now deﬁne the groupoid Lip Met∗ with objects Lip X for each metric space X, with distance function

d (Lip X, Lip Y ) := inf ǫ 0 : κ : Lip Y Lip X ǫ-ml-isom. ml { ≥ ∃ → } modulo pseudo-ml-isometry dml = 0. We endow Lip Met∗ with rough ml- isomorphisms as morphisms. In these terms, the mapping ¯ : η η¯ is a · 7→ Lipschitz equivalence between the metric categories Met∗ and Lip Met∗, and a contravariant functor up to nearness of rough isometries. Unfortunately, we are not yet able to generalize Theorem 101 to ǫ-short maps and ǫ-ml-short maps, which would be the appropriate morphisms of Met and Lip Met (see Section 0.1.2). 70 Claim 118 Chapter 3

Rough Isometries of Groups

3.1 The Theorem of Mazur-Ulam

Some Theorems and Lemmas have the property of being stable against perturbations of their input. We want to give an example for this in form of a variant of the Banach Fixed Point Theorem. For this, we copy the standard proof from [Ho] and replace all steps in the proof by their rough counterparts: A point is replaced by a ball, uniqueness is replaced by bounded distance, and so on.

Theorem 119 119 Let M be a non-empty true metric space (not necessarily complete), and T : M M such that there are 0 q < 1 and ǫ 0 with → ≤ ≥ d(Tx, Ty) q d(x, y) + ǫ ≤ for all x, y M. Then for each R>ǫ (2 q)/(1 q)2 there is a point x M ∈ − − 0 ∈ such that T B (x ) B (x ), and any two such points are within distance R 0 ⊆ R 0 (2R + ǫ)/(1 q). ≤ −

Proof Let r := R (1 q) ǫ > ǫ/(1 q). By iteration we see that − − − n n n 2 n 1 d(T x, T y) q d(x, y) + (1+ q + q + . . . + q − ) ǫ ≤ · qn d(x, y) + ǫ/(1 q) ≤ − holds for all n N . Let x M be arbitrary, then follows ∈ 0 ∈ n n 1 n 1 d(T x, T − x) q − d(Tx, x) + ǫ/(1 q). ≤ − n N As q 0 for n , there is N N and x0 := T x with → →∞ ∈ ∗ d(T x , x ) r 0 0 ≤ (This is the rough counterpart of the Cauchy criterion.) For all y B (x ) ∈ R 0 d(Ty, x ) d(Ty,Tx ) + d(T x , x ) q R + r + ǫ = R 0 ≤ 0 0 0 ≤ 72 Claim 120 holds, hence T B (x ) B (x ). Now let y M be another point with R 0 ⊆ R 0 0 ∈ T B (y ) B (y ). Then R 0 ⊆ R 0 d(x , y ) d(x ,Tx ) + d(T x ,Ty ) + d(Ty , y ) 0 0 ≤ 0 0 0 0 0 0 2 R + q d(x , y ) + ǫ ≤ 0 0 d(x , y ) (2 R + ǫ)/(1 q). ⇒ 0 0 ≤ −

This procedure works for all proofs of sufficient simplicity, which are straight- forward applications of (in)equalities or quantifiers. For example, each of the intervaluation laws derivable by a finite Venn diagram (see 64) still roughly holds when the cut law of Definition 62 is replaced by w(f, g h) w(f h, g) ǫ w(f, g) w(f, g h) + w(f h, g) + ǫ. ∨ ◦w ∧ − ≤ ≤ ∨ ∧

Unfortunately, not every proof can be reformulated in a rough context. Here, we will first quote a very elegant proof of the Mazur-Ulam Theorem ([MU]), as given by Väisäläin [V]. We then conjecture a rough version of the Mazur-Ulam Theorem, and show how Väisälä’s proof fails to adapt to the rough context. (Note in comparison that with an “isometry” we always mean a bijective isometry.)

Theorem 120 (Mazur-Ulam) 120 Every isometry f : E F between normed ﬁnite-dimensional vector spaces is → aﬃne (i.e. linear plus constant).

Proof We quote from [V]: Let a, b E be arbitrary, and put z := (a + b)/2. ∈ Let W be the family of all isometries with fixed points a and b. Let λ := sup g(z) z : g W . As a is a fixed point of each g W , we have {|| − || ∈ } ∈ g(z) a = g(z) g(a) = z a , and || − || || − || || − || g(z) z g(z) a + a z = 2 a z , || − || ≤ || − || || − || · || − || hence λ is finite. Let ψ be the reflection of E in z, this is ψ : x 2 z x. For each g W 7→ − ∈ holds g = ψ g 1 ψ g W as well, as ψ maps a and b onto each other. ψ is an ∗ − ∈ isometry, and its only fixed point is z. This implies

ψ(x) z = x z − − and ψ(x) x = 2 x z . − − 1 Now, g and g− are isometries, so we ﬁnd

1 2 g(z) z = ψ(g(z)) g(z) = (g− ψ g)(z) z − 1 − − = (ψ g− ψ g)(z) z = g∗(z) z λ. − − ≤

The Theorem of Mazur-Ulam 73

For each δ > 0 we may choose g W with g(z) z λ δ, which yields ∈ || − || ≥ − 2 λ λ + δ, consider δ 0 and ﬁnd λ = 0, thus g(z) = z whenever g is an ≤ → isometry with ﬁxed points a and b.

Now let f : E F be any isometry, and let z′ := (f(a) + f(b))/2. Let ψ′ → 1 be the reflection at z′, then h := ψf − ψ′ f fixes a and b (a is first mapped to f(a), then to f(b), to b, and finally back to a by ψ), hence h(z) = z. But this 1 1 means (ψf − ψ′ f)(z) = z, and, as ψ− (z) = z, simply ψ′(f(z)) = f(z). As ψ′ is a reflection, there is only one fixed point, and this is z′; hence, we have z′ = f(z), and thereby a + b f(a) + f(b) f = . 2 2 Now define g(x) := f(x) f(0). From direct calculation follows − x + y g(x) + g(y) g = . 2 2 for all x, y E. Insert y = 0 to find g(2x) = 2 g(x), and subsequently ∈ 1 1 g(x + y) = g(2 x) + g(2 y) = g(x) + g(y). 2 2 Furthermore, we have

g 2j x = 2j g(x)  ·  · j J j J X∈ X∈   for any ﬁnite J Z. Continuity of g yields its full linearity. ⊆

Conjecture 121 (Mazur-Ulam, rough version) ...... 121 Every ǫ-isometry f : E F between normed finite-dimensional vector spaces → is O(ǫ)-affine (affine up to an additive error which is a multiple of ǫ).

Non-Proof Let a, b E be arbitrary, and z := (a + b)/2. For any δ > 0 ∈ deﬁne W to be the family of all ǫ-isometries g, such that d(g(a), a) δ and δ ≤ d(g(b), b) δ. Let λ := sup g(z) z : g W . Similar to the original ≤ δ {|| − || ∈ δ} proof, we have

d g(z) a , z a 2 δ − − ≤ and g(z) z g(z) a + a z 2 a z + 2 δ. − ≤ − − ≤ − Again, this implies λ is finite, with a bound which depends on a b . However, δ || − || this would not suffice for the final conclusion, as we have to apply this for all a, b E; what we need to show is that λ has a bound indepent of a b . ∈ δ || − || 74 Claim 121

Let ψ be the reflection x 2 z x of E in z. For each g W with rough inverse 7→ − ∈ δ g W holds g := ψ g ψ g W , as g(a) a δ, (ψ g)(a) b δ, ′ ∈ δ ∗ ′ ∈ 3δ || − || ≤ || − || ≤ (g ψ g)(a) b 3 δ, (ψ g ψ g)(a) a 3 δ (same for b). In contrast to || ′ − || ≤ || ′ − || ≤ g and g , ψ still is an isometry, with fixed point z, ψ(x) z = x z , and ∗ || − || || − || ψ(x) x = 2 x z . || − || || − || g and its rough inverse g′ are δ-isometries, so we find

2 g(z) z = ψ(g(z)) g(z) − − (g′ ψ g)(z) z + 2 δ ≤ − = (ψ g′ ψ g)(z) z + 2 δ − = g∗(z) z + 2 δ λ3δ + 2 δ − ≤

Choosing an appropriate sequence of g W , and including the already known j ∈ δ bound for λ, we ﬁnd the following two restrictions:

1 λ A + 2 δ and λ λ + δ δ ≤ δ ≤ 2 3δ with A := 2 a z = a b . || − || || − || We now give an example to show that these restrictions are not strong enough to prove that λδ has an upper bound independent of A: Set

4 λ (A) := 2 √A δ3 . δ · · Then due to √4 Aδ √δ 2 0 and √A √δ 2 0 we have − ≥ − ≥ λ (A) √A δ + δ δ ≤ 1 3 A + δ A + 2 δ. ≤ 2 2 ≤

On the other hand, λ = λ √4 27 λ √4 16 2 λ . 3δ δ · ≥ δ · ≥ δ ×

The first equations and inequalities of our “proof” started out well. It began to wallow just in the moment we introduced Wδ: We categorized certain isometries, and in contrast to the prior proof, it was not possible to categorize g∗ the same way, it landed in W3δ. Why did we have to use Wδ the way we defined above? 1 In the original proof, the connection to the isometry f was that h := ψf − ψ′ f would be in W . In our case, with f an ǫ-isometry, h would be a 2ǫ-isometry, and hence we had to define W in one of two ways: Either the way we chose above (and failed), or to allow any rough isometry which approximately fixes a and b to enter W . This choice would have broken down the moment we try to prove that λ is finite, as δ might have been arbitrarily large. In retrospective, it seems plausible that the second-order-logic Väisäläapplied is the obstacle against roughification of the proof, and that it might be possible to prove that any first-order-logic proof is stable against rough perturbations. Coarse Relations for Groups 75

3.1.1 Rough Abelianness

Apart from functions and theorems, it is also possible to replace axioms by their rough counterparts. We only want to touch upon this theme by giving a short categorization for rough abelianness.

Example 122 ...... 122 Let Γ be some finitely generated group, d some word metric on G and ǫ > 0 fixed. Let us assume furthermore, that for all g, h Γ holds: ∈ d(gh, hg) ǫ. ≤ As Γ is finitely generated, we have: d(gh, hg) ǫ for all g, h Γ for some ≤ ∈ ǫ > 0 if and only if the set of commutators in Γ is finite. According to [Ba], this is in turn equivalent to the commutator group [G, G] being finite, and hence the abelianized group Gab = G/[G, G] being of finite index. Our forthcoming Propositions 134 and 135 will then yield a natural δ-isometry between G and Gab with δ = 1 + diam([G, G]). Note that the diameter of [G, G] might be larger than ǫ. Conclusion: To be abelian is a roughly stable property; each roughly abelian group is roughly isometric to an abelian group (its abelianization).

3.2 Coarse Relations for Groups

When one speaks about the coarse geometry of finitely generated groups, one generally means quasi-isometries of Cayley graphs. While a single infinite group gives rise to an infinite number of non-isomorphic Cayley graphs, quasi- isometries do not depend on the generating system of the group, and hence the quasi-isometry class of a group is well-defined, and an important invariant. It encompasses the idea of two groups being approximately isomorphic, but quasi- isometries are not the only way to do this. Particularly the pure group-theoretic notion of commensurability rivals the quasi-isometry, and their interplay is still an interesting research problem. In the following, we use the definitions given in [dH].

Definition 123 ...... 123 Let G and H be groups. G and H are commensurable when there exist subgroups G G and H H of finite index, such that G and H are isomorphic as ′ ≤ ′ ≤ ′ ′ group. G and H are commensurable up to finite kernels if there exists a finite sequence of groups Γ1, ..., ΓN and homomorphisms h0,...,hN

h h h h hN−1 h G 0 Γ 1 Γ 2 Γ 3 . . . Γ N H −→ 1 ←− 2 −→ 3 ←− −→ N ←− with ﬁnite kernels and images of ﬁnite index. 76 Claim 126

One easily sees that commensurability always implies commensurability up to ﬁnite kernels, which in turn always implies quasi-isometry, given that both groups are ﬁnitely generated. We quote without proof the following Proposition from [dH], IV.28.

Proposition 124 ...... 124 Two residually ﬁnite groups are commensurable if and only if they are commensurable up to ﬁnite kernels.

Proposition 125 ...... 125 Let G and H be f.g. groups, and η : G H a homomorphism and quasi- → isometry. Then G and H are commensurable up to ﬁnite kernels.

Proof The kernel of φ is finite, because it is the preimage of a finite subset of H. And the image φ(G) is a subgroup of H of finite index: φ(G) is ǫ-dense in H. Let B be the ǫ-ball around the identity in H, then each element h H ∈ can be written as b φ(g) for some b B and g G. With this, the number of · ∈ ∈ cosets of G/φ(g) can be at most as large as #B, and in particular, it is finite.

There is a multitude of cases in which quasi-isometry implies commensurability (for example f.g. abelian groups, certain types of Baumslag-Solitar groups, abelian-by-cyclic groups in [FM]) but also a plenty supply of counter-examples (e.g. Lamplighter groups, or Z2 ⋊ Z with certain choices for A GL(2, Z)). A ∈ We want to add to this discussion by deﬁning new kinds of coarse equivalences for f.g. groups, situated between commensurability and quasi-isometry, and mostly based upon rough isometries of Cayley graphs. Rough isometries are stronger than quasi-isometries, and hence we might expect equivalence to commensurability for larger classes of groups. However, rough isometries are not a canonical notion for groups, as they depend on the chosen generating set. This problem unfolds into a rich zoo of diﬀerent notions of rough isometry for groups.

Example 126 ...... 126 The weakest of these notions motivates a theorem of the form:

Groups are commensurable if and only if they admit generating systems such that their Cayley graphs are roughly isometric.

Unfortunately, this is wrong, the property is too weak. The counter-example are the lamplighter groups (see [dH] IV.44)

Γ := F ⋊ Z F  j j Z M∈   where Z acts by shifting, and each Fj is a copy of a ﬁnite group F . For ﬁnite groups F , G of same size there are generating systems of ΓF and ΓG such that Exponential Growth Rate 77 the corresponding Cayley graphs are even isomorphic as graphs. Now one may choose F solvable and G not. Then ΓF will be solvable, but ΓG not virtually solvable, which implies that they cannot be commensurable. This is the classic example to show that quasi-isometry does not imply commensurability, and it works for rough isometries (and even isometries) equally well.

A related question is whether two isomorphic Cayleygraphs imply that their groups are isomorphic. This is in general not the case (as long as only one Cayleygraph per group is considered), and is already in the finite case a rich source for problems, see [L]. We are not yet able to categorize all of these notions and give proofs or counter- examples to their mutual equivalences. However, we will take a closer look at two of these definitions. Our methods involve the analysis of generating systems and groups of what we call “shared isometries” and “shared rough isometries” – maps which are isometries (respectively rough isometries) relative to lots of generating systems at once. But before we get there, we insert a section about a rough isometry invariant, and another section about the case in abelian groups. These two sections together provide first insights.

3.3 Exponential Growth Rate

Each quasi-isometry invariant is also a rough-isometry invariant. But there also is a rough isometry invariant, which is not a quasi-isometry invariant:

Deﬁnition 127 ...... 127 Let G be a f.g. group, and let S be a ﬁnite generating system of G. The exponential growth rate is

k ln #BG,S(k) ω(G, S) := lim sup #BG,S(k) = exp limsup . k k k →∞ q →∞ The minimal growth rate ω(G) is the inﬁmum of ω(G, S) over all ﬁnite generating systems S. The group G is of uniformly exponential growth if ω(G) > 1.

Proposition 128 ...... 128 Let F be the free group on n generators. Then ω(F ) = 2 n 1. n n −

Proof This is Proposition VII.12 in [dH], we summarize the proof here: The minimal growth rate is attained by any free generating system for Fn. Now let S be any generating system of Fn. Let S′ be the image of S under abelianiza- n tion, choose a minimal subset of S′ generating a ﬁnite index subset of Z . Any preimage of S′ is a set of free generators of a subgroup H of Fn, which in turn is isomorphic to Fn and of growth 2 n 1. Hence, ω(Fn, S) ω(H, S′) = 2 n 1. − ≥ − 78 Claim 131

Lemma 129 ...... 129 Let G, H be f.g. groups of uniformly exponential growth, let SG and SH be ﬁnite generating systems of G and H, and let

η : Cay(G, S ) Cay(H, S ) G → H be an ǫ-isometry, ǫ 0. Then ω(G, S ) = ω(H, S ). ≥ G H Proof By estimating the number of elements in each ball:

#BG(r) #η(BG(r)) #BG(ǫ) ≤ #B (r + ǫ) ≤ H #B (r) #B (ǫ) ≤ H · H ω(G, S ) ω(H, S ), ⇒ G ≤ H and vice versa.

Example 130 ...... 130 Although the free groups F2 and F4 are commensurable, there is a generating system of F2, such that its Cayley graph is not roughly isometric to any Cayley graph of F4, as the minimal growth rates diﬀer.

However, there still exist generating systems of F2 and F4 with their Cayley graphs being roughly isometric: Choose any embedding π of F4 into F2 as subgroup of ﬁnite index, let Sj be free generating systems of Fj, j = 2, 4. Choose S := S π(S ) as generating system for F , then due to the uniqueness of 2 ∪ 4 2 each word in F2 and due to the corresponding unique length function, π is a rough isometry between Cay(F2, S) and Cay(F4, S4).

3.4 The Abelian Case

Lemma 131 ...... 131 Let N be a ﬁnite subgroup of GL(n, Z), n N , G = Zn, and G = G ⋊ ∈ ∗ 0 0 N with canonical action. Then for each generating system S of G there is a n n generating system S′ of Z , such that the canonical embedding i : Z ֒ G is a n → rough isometry of Cay(Z , S′) and Cay(G, S).

Proof Choose

1 S′ := g− s g g N, s S . { | ∈ ∈ } Let d be the metric in Cay(Zn, S ), and d the metric in Cay(G, S). Let x G ′ ′ ∈ 0 be arbitrary. As S S , we obviously have d (0, x) d(0, x). Now represent ⊆ ′ ′ ≤ x in S′: x = sg1 sg2 . . . sgk 1 · 2 · · k The Abelian Case 79

A B

C D

1 0 1 0 1 1 0 1 A = B = C = − D = 0 1 0 1 0 1 1 1 − − − −

Figure 3.1: A collection of balls of radius 12 in various groups of kind G = Z2 ⋊ M , with M GL(2, Z), projected on the canonical subgroup h i ∈ Z2. All four of these Cayleygraphs are constructed from the same generating system, which is depicted in the upper image. While the trivial case A is the expected convex span of the generating system (approximately, and modulo the discrete structure of Z2), non-trivial matrices generate larger balls by enforcing approximate symmetries by their action: If w is a word in Z2 of length L, then Mw is of length L +1 in G. ≤ 80 Claim 132 with g N, s S, and g g ...g = e, because x G . Then by commu- j ∈ j ∈ 1 2 k ∈ 0 tativity we can rearrange the word to collect all instances of an element of N:

g x = sg,1 sg,2 ... sg,l(g) g N Y∈ with l(g) = k. Represent each of the ﬁnitely many g N with a minimal ∈ word w of letters in S. Let L be the greatest length among the w , then x is Pg g of length k + ǫ with ǫ = 2 #N L. Hence i is an ǫ-isometric embedding. ≤ · · Now let x G be arbitrary. As N is a normal subgroup, there is g N and ∈ ∈ h G such that x = g h. The element g is of length L in S, hence G is ∈ 0 ≤ 0 L-dense in G, and i is an ǫ-isometry.

Corollary 132 ...... 132 n n Let G = Z ⋊ N be generated by a generating system S′ of Z and the whole of N r e . Let r N be arbitrary, then the r-ball in G is at bounded Hausdorﬀ- { } ∈ ∗ distance to the set A Zn which is constructed in the following way: ⊆ n 1. Take A1 to be the r-ball of S′ in Z ,

n 2. A2 the canonical embedding of A1 into R ,

3. A3 the union of the orbit of A2 under N,

4. A4 the convex hull of A3,

5. and ﬁnally A = A Zn. 4 ∩ We may restrict any word metric on G to its subgroup Zn and visually compare the possible geometries by comparing their generated unit balls in Zn, N will then impose symmetries on the possible geometries. This can be seen in Figure 3.1 for the case n = 2 and three choices of cyclic subgroups of GL(2, Z).

While the quasi-isometry and commensurability classes of a group are given by any (normal) subgroup of ﬁnite index, the ﬁnite quotients can still modulate the possible rough isometry classes of a group:

In the above case of semidirect products of abelian groups, they simply • restrict to metrics suitable for the corresponding symmetries: The group has less or equally many rough isometry classes than its subgroups.

In the case of free groups in section 3.3, the ﬁnite quotients may as well • increase the number of possible metrics, as the exponential growth rate shows (F2 allows metrics which cannot be generated by its ﬁnite-index subgroup F4): Here, the group has more or equally many rough isometry classes than its subgroups. Rough Isometries of Quotients with Finite Kernel 81

Example 133 ...... 133 We now give an example that the relation

G H : S , S generating systems, such that Cay(G, S ) ∼ ⇔ ∃ G H G and Cay(H, SH ) are roughly isometric is not an equivalence relation per se. We choose

0 1 0 1 G := Z2 ⋊ and H := Z2 ⋊ . 1 1 1 0 − − By Lemma 131 we have G Z2 H. Assume there are generating systems ∼ ∼ SG and SH such that Cay(G, SG) and Cay(H, SH ) are roughly isometric. Each r-ball must (approximately) adhere to both symmetry groups. The order of the ﬁrst one is 6, the order of the second is 4, but there is no ﬁnite subgroup of GL(2, Z) of order lcm(6, 4) = 12 or higher.

3.5 Rough Isometries of Quotients with Finite Kernel

Proposition 134 ...... 134 Let G be a f.g. group, let H E G be a finite normal subgroup, and set G′ := G/H. Then each finite generating system S0 of G can be enlarged to a finite generating system S of G, such that η : G G , g gH is a 1-isometry, where → ′ 7→ G′ is endowed with the word metric of the projection S′ of S.

Proof Let S be some generating set of G, and put S := S H r e . Deﬁne 0 0 ∪ { } S := sH : s S r e as the non-trivial cosets of S. S generates G : For ′ { ∈ } { } ′ ′ each x G is x = gH for some g G, present g as s ...s with s S . ∈ ′ ∈ 1 n j ∈ 0 Then x = (s H) . . . (s H). From this we see d (eH, gH) d(e, g) with d 1 · · n ′ ≤ the word metric resulting from S G and d the word metric for S G . ⊆ ′ ′ ⊆ ′ On the other hand, let g G be arbitrary, and let gH = (s H) . . . (s H) G ∈ 1 · · n ∈ ′ be a shortest word in G . Then there is h H with g = s ...s h, hence ′ ∈ 1 n · d(e, g) d (eH, gH) + 1. Finally, let x G be any coset. Choose any rep- ≤ ′ ∈ ′ resentative g of this coset, thus gH = x. Then d(x, η(g)) = d(x, gH) = 0.

Note that in the preceding proof we might have chosen some generating set SH of H and set S := S S . In this case, the proof would yield an ǫ-isometry 0 ∪ H with ǫ = diam Cay(H,SH ) instead.

Proposition 135 ...... 135 Let S be some generating set of the ﬁnitely generated group G, H E G ﬁnite, and SH a generating set of H. Then the identity (G, dS) (G, dS SH ) is an → ∪ ǫ-isometry with ǫ diam (H). ≤ S 82 Claim 136

Proof Let d := dS, d′ := dS H . We obviously have d′(e, g) dS SH (e, g) ∪ ≤ ∪ ≤ d(e, g) for all g G. Now let g = s t ...s t be some presentation of g G ∈ 1 1 n n ∈ in generators s S e and t H. As H is normal, we can ﬁnd t to j ∈ ∪ { } j ∈ 1′ t H with g = s ...s t ...t . Hence d(e, g) d (e, g) + ǫ where ǫ is n′ ∈ 1 n · 1′ n′ ≤ ′ the diameter of H G in d . ⊆ S

3.6 Shared Isometries

Definition 136 ...... 136 Consider ǫ 0, and let G be a finitely generated group. Let S be a family of ≥ generating systems. Define the S-shared or simply shared isometry groups and sets

(λ, ǫ) -IsomS(G) := η : G G S S : η is a (λ, ǫ)-qi. rel. to S { → | ∀ ∈ } ǫ -IsomS(G) := (1, ǫ) -IsomS(G) IsomS(G) := 0 -IsomS(G)

UQIsomS(G) := (λ, ǫ) -IsomS(G) λ,ǫ 0 [≥ URIsomS(G) := ǫ -IsomS(G) ǫ 0 [≥ The last ones we call S-uniform quasi-isometries resp. rough isometries. We further deﬁne

ǫ -IdenS(G) := η : G G S S : η is ǫ-near the identity { → | ∀ ∈ } IdenS(G) := ǫ -IdenS(G). ǫ 0 [≥ These definitions are similar to the definition of the quasi-isometry group QI of a true metric space or group (the calculation of QI is very difficult in general, see for example [FM]), and we find composition to be a group structure on UQIsomS(G) and on URIsomS(G) after quotiening out IdenS(G). The difference between the quasi-isometry group QI(G) and UQIsomS(G)/ IdenS(G) seems to be subtle, as we just demand λ and ǫ to be uniformly bounded for all word metrics in S, but this difference can be enormous, if S is chosen large enough. On the other hand, if S comprises only a finite number of generating systems, UQIsomS(G)/ IdenS(G) equals QI(G), independently of the exact choice of S. We will begin with the examination of UQIsomS(G) and URIsomS(G) in Section 3.7, and now concentrate on the nearly trivial case of IsomS(G). We start with a simple observation, which resulted from a discussion with Laurent Bartholdi and Martin Bridson during the 2007 winter school “Geometric Group Theory” in Göttingen:

Theorem 137 (L. Bartholdi ’07) 137 (A) Let S = Sasym be the family of all, possibly asymmetric, ﬁnite generating Shared Isometries 83

systems of G. Then IsomS(G) is isomorphic to G (using possibly asymmetric distance functions).

(B) Let G be a group with a ﬁnite, symmetric generating system S0 such that the following hold:

1. There are no s ,s ,s S with s s = s . (Minimality; easy to achieve.) 1 2 3 ∈ 0 1 2 3 1 2 2 2. There are no s ,s S , s = s± , with s s = e. 1 2 ∈ 0 1 6 2 1 2 1 s2 1 3. There are no s ,s S , s = s± , with s = s− . 1 2 ∈ 0 1 6 2 1 1 1 s2 4. There are no s ,s S , s = s± , with s = s 1 2 ∈ 0 1 6 2 1 1 (In particular, G is not an abelian group.)

5. There are at least two distinct elements in S0, which are not inverses of each other.

Let S = Ssym be the family of all symmetric ﬁnite generating systems of G. Then IsomS(G) is isomorphic to G.

(C) Let G be a f.g. abelian group without 2-torsion, and let S = Ssym be the family of all symmetric ﬁnite generating systems of G. Then IsomS(G) is isomorphic to G ⋊ C , where C acts by inversion x x 1. 2 2 7→ − (D) Let G be a f.g. group, and S S = S (G), such that S is minimal, 0 ∈ sym 0 and each element s S has order 2 (i.e. s2 = e). Then IsomS(G) = G. ∈ ∼

Proof (A) Consider φ IsomS(G), and x,s G arbitrary, s = e. Let S := ∈ ∈ 6 ′ S S : s S . Then d (x,xs) = 1 and d (φ(x), φ(xs)) = 1 for each S S , { ∈ ∈ } S S ∈ ′ i.e. s := φ(x) 1 φ(xs) S. Assume s = s. Then deﬁne S := (S r s ) x − · ∈ x 6 ′ { x} ∪ s, s 1s . S is again a generating system and s / S , as s = s and s = e. { − x} ′ x ∈ ′ 6 x 6 Yet, we have s S , contradiction. So we conclude s = s and φ(xs)= φ(x) s. ∈ ′ x · By induction we ﬁnd φ(x) = φ(e) x, with φ(e) arbitrary. On the other hand, · each such φ obviously is in IsomS(G), and

G g (φ : x g x) IsomS(G) ∋ 7→ g 7→ · ∈ are shared isometries, and φ φ = φ . g ◦ h gh (B) Let φ IsomS(G), and x G arbitrary, s S . Then d (x,xs) = 1 and ∈ ∈ ∈ 0 S0 d (φ(x), φ(xs)) = 1, i.e. s := φ(x) 1 φ(xs) S . Like in the asymmetric case, S0 x − · ∈ 0 using S := S S : s S S we find s = s or s = s 1, but the choice ′ { ∈ ∈ } ∋ 0 x x − might depend on x, and this is the main point differing to the asymmetric case. Now let r S be arbitrary, r = s 1 and S := S sr, (sr) 1 . Note that ∈ 0 6 ± 0′ 0 ∪ { − } d (x,xsr) = 2, as there are no triangles in S , but d ′ (x,xsr) = 1. Let r = S0 0 S0 y 1 φ(y)− φ(yr) S0, so we find φ(xsr)= φ(x) sx rxs. As d ′ (φ(x), φ(xsr)) = 1, · ∈ · · S0 we have

1 1. sx = s or sx = s− ,

1 2. rxs = r or rxs = r− , 84 Claim 137

3. s r S , but s r / S . x xs ∈ 0′ x xs ∈ 0 1 1 1 Hence, sxrxs must be one of the added elements sr or (sr)− = r− s− . We ﬁnd eight cases:

1. sx = s, rxs = r, sxrxs = sr 2. s = s 1, r = r, s r = sr s2 = e y case (1) x − xs x xs ⇒ 3. s = s, r = r 1, s r = sr r2 = e y case (1) x xs − x xs ⇒ 4. s = s 1, r = r 1, s r = sr s2r2 = e x − xs − x xs ⇒ 5. s = s, r = r, s r = r 1s 1 (sr)2 = e y case (1) x xs x xs − − ⇒ 6. s = s 1, r = r, s r = r 1s 1 rs = r 1 x − xs x xs − − ⇒ − 7. s = s, r = r 1, s r = r 1s 1 sr = s 1 x xs − x xs − − ⇒ − 8. s = s 1, r = r 1, s r = r 1s 1 rs = r x − xs − x xs − − ⇒ Cases (2), (3) and (5) directly lead to case (1) after re-inserting, case (4) contradicts property (2) for S0, cases (6), (7) and (8) contradict properties (3) and (4). Hence, we are left with case (1), and s = s for all x G. Again, we use x ∈ induction to show φ(x)= φ(e) x, and get an isomorphism · G g (φ : x g x) IsomS(G) ∋ 7→ g 7→ · ∈

(C) It is easy to find a generating system S0 of G which fulfills all properties s2 of subtheorem (B), except for property (4): s1 = s1 is always true. We follow through the proof of subtheorem (B) until case (8) cannot be contradicted. Assume it is realized, i.e. we find x G, s S with φ(xs)= φ(x) s 1. Then, for ∈ ∈ 0 · − each r S r s,s 1 we must have φ(xsr)= φ(x) s 1 r 1, and from excluding ∈ { − } · − · − all other cases and property (2) of S we further find φ(xs2) = φ(x) s 2. By 0 · − induction and using the fact that S0 generates G, we show

1 1 1 φ(s s ... s ) = φ(e) s− s− ... s− , 1 2 n · 1 2 n or, due to abelianness, φ(x) = φ(e) x 1. Obviously, all these bijections are · − indeed shared isometries:

1 1 1 d(φ(x), φ(y)) = d(x− , y− ) = xy− abelianness 1 || || | = y− x = d(y, x) S is symmetric || || | 0 = d(x, y)

Hence, we have IsomS(G) isomorphic to G ⋊ C2 via

a G ⋊ C (g, a) (φ : x g x ) IsomS(G) 2 ∋ 7→ (g, a) 7→ · ∈

(D) Once again, we follow through the proof of subtheorem (B). As S0 is minimal, property (1) is automatically fulﬁlled. And as each s S has order 2, ∈ 0 Shared Isometries 85

1 1 the question sx = s or sx = s− is trivial, as s− = s. Hence, we get the usual isomorphism

G g (φ : x g x) IsomS(G). ∋ 7→ g 7→ · ∈

From now on, we will restrict to the symmetric case S = Ssym.

Example 138 ...... 138 For groups G with central elements it can be diﬃcult to ﬁnd a generating system S0 satisfying the properties of Theorem 137.B, but typically it is still possible. Take for example:

G = a, b, c [a, c], [b, c] ∼= (Z Z) Z h 1 | 1 i1 ∗ × S0 = a± , (bc)± , (ab)± . Example 139 ...... 139 The same accounts for groups with 2-torsion. For example, it is easy to calculate by hand

IsomS(C2) ∼= C2, just as Theorem 137.D mentions; but not C2 ⋊ C2, as one might think from Theorem 137.C. Indeed, as inversion is the trivial operation in each group of n n exponent 2, we have IsomS(C2) ∼= (C2) in the abelian case, contrary to The- orem 137.C.

Example 140 ...... 140 For groups of the form G = G0 ⋊ C2 with C2 acting via inversion (written multiplicatively) on a f.g. group G0 (which subsequently must be abelian), each element (g, 1) with g G has torsion 2. Given a minimal generating system − ∈ 0 S0 of G0, we can use

S := (g, 1) : g S (e, 1) { − ∈ 0} ∪ { − } to apply Theorem 137.D. And, just as it states, the inversion is not a shared isometry in this case: Let G be any f.g. group with at least one element s G 0 ∈ 0 with s2 = e, S a ﬁnite generating system of G with s S , and S := S 6 0 0 ∈ 0 ′ 0 ∪ (s, 1) , which generates G = G ⋊ C . Then holds d (e, 1), (s, 1) = 1, { − } 0 2 − as (e, 1) (s, 1) = (s, 1), but − · − 1 1 1 d (e, 1)− , (s, 1)− = d (e, 1), (s− , 1) > 1, − − because (s 1, 1) / S . (s = s 1, and (s, 1) 1 = (s, 1).) − − ∈ ′ 6 − − − − 86 Claim 142

Example 141 ...... 141 Similar to Example 140, consider a group G = G0 ⋊ H, where a f.g. group H acts on the f.g. abelian group G0. The action shall be given by a non-trivial homomorphism α : H C , where C acts on G by inversion. Furthermore, → 2 2 0 let S0 be an arbitrary finite generating system of G0, and let SH be a finite generating system for H, such that there are no two elements s, t S with ∈ s = t 1 and st = s 1 or s2 t2 = e. Finally, let h S be an element with 6 ± ± 0 ∈ H h4 = e. Then we can define a finite generating system 0 6 S := S gh : g S 0 H ∪ 0 ∈ 0 from which we choose a minimal subsystem S S . Some simple calculations ⊆ 0 then show that the generating system S fulfills the requirements for Theorem 137.B, and we conclude:

IsomS(G0 ⋊ H) ∼= G0 ⋊ H In particular, this accounts for the group

y 1 2 2 Z⋊Z = x, y : x = x− = y, z : y = z . h i ∼ h i

Considering the proof of Theorem 137 and the above examples, we are conﬁdent that the following statements can be proven just by application of more arduous combinatorics:

(A) Let G be a f.g. group, and let S be the family of all symmetric generating systems of G. Then IsomS(G) ∼= G ⋊ C2 if and only if G is non-trivial, abelian, and not of exponent 2; IsomS(G) ∼= G otherwise. (B) The shared Cliﬀord isometries (i.e. those shared isometries φ with constant d(x, φ(x)) for all x G) always constitute a group, ∈ which is isomorphic to G.

Lemma 142 ...... 142 Let G, H be f.g. groups, SG, SH families of generating systems of G, H. If there is a bijection η : G H such that → for each S S there is S S which makes η : Cay(G, S ) • G ∈ G H ∈ H G → Cay(H, SH ) an isometry, and

for each S S there is S S which makes η 1 : Cay(H, S ) • H ∈ H G ∈ G − H → Cay(G, SG) an isometry.

Then IsomSG (G) and IsomSH (H) are isomorphic. In particular, in the situations of Theorem 137.A, B, or D, or when G and H are both f.g. abelian without 2-torsion (case (C)), then G and H are isomorphic. Shared Rough and Quasi-Isometries 87

Proof Deﬁne

η∗ : IsomSG (G) IsomSH (H) → 1 φ η φ η− . 7→ ◦ ◦ This is well-deﬁned: For each S S choose S S such that η is an H ∈ H G ∈ G isometry. Then η φ η 1 : H H is an isometry as well—vice versa for ◦ ◦ − → (η ) 1 := η 1 η. Hence, η is a bijection, and, as one easily computes, ∗ − − ◦ · ◦ ∗ indeed an isomorphism between groups.

In the cases (A), (B) and (D), we may directly conclude G ∼= H. In the abelian case we just have G ⋊ C2 ∼= H ⋊ C2, but, as G and H are without 2-torsion, G and H must be isomorphic as well.

Example 143 ...... 143 Let us take a look at the three commensurable groups G1 = Z, G2 = Z⋊ C2 ∼= C C , and G = Z C . For G , choose S = 1, 2 , and apply Theorem 2 ∗ 2 3 × 2 1 0 {± ± } 137.C; for G2 use Theorem 137.D (c.f. previous example); for G3 apply a direct calculation1. Then we ﬁnd

IsomS(G1) ∼= Z⋊ C2 IsomS(G2) ∼= Z⋊ C2 IsomS(G ) = (Z⋊ C ) C = G ⋊ C . 3 ∼ 2 × 2 ∼ 3 2 We note that the resulting shared-isometry groups can be isomorphic, but might as well be just commensurable. And, as G1 and G2 are not isomorphic, we note that there cannot be a bijection η : G G as in Lemma 142. The canonical 1 → 2 inclusion i : G ֒ G however might provide a deeper insight - it is a rough 1 → 2 isometry for several generating systems.

3.7 Shared Rough and Quasi-Isometries

We have seen in Section 3.1 that it is sometimes possible to directly translate a proof into the rough context. This will be our goal for this section: To roughiﬁcate the proof of Theorem 137.

Deﬁnition 144 ...... 144 Let G be a f.g. group. We call a family S of ﬁnite generating systems of G optimal if URIsomS(G) ∼= G, and quasi-optimal if UQIsomS(G) ∼= G.

1In this case it suﬃces to ﬁnd the isometries for the standard generating set, the Cayley graph of which is a ladder. The cardinality of the second neighborhood of an edge in this graph depends on the order of its generating element, but must be preserved under isometries. This allows for a simple case distinction. 88 Claim 145

Note that quasi-optimality is the stronger of both notions, as URIsomS(G) ⊆ UQIsomS(G). Each translation from the left with an element of G is a shared isometry, and hence we have

G IsomS(G) URIsomS(G) UQIsomS(G). ≤ ⊆ ⊆ If S is optimal, we also ﬁnd IdenS(G) to be trivial.

Lemma 145 (Optimality Lemma) ...... 145 y 1 Let G be a ﬁnitely generated group with x = x− for all x, y G, unless 1 6 ∈ x = x− . Let G be non-abelian, or of exponent 2. Let S be a family of ﬁnite generating systems of G with the following Property 145:

For each g, h G with g = h 1 and each R N there is • ∈ 6 ± ∈ ∗ S = S(g, h, R) S such that g S and h R, or vice versa. ∈ ∈ || ||S ≥ Then S is quasi-optimal (and thus optimal).

Proof Let λ, ǫ 0, φ (λ, ǫ) -IsomS(G), and x, y G be arbitrary, let ≥ ∈ ∈ z := y 1 x and deﬁne − · 1 z′ := φ(y)− φ(x) z′ = d (φ(y), φ(x)). · ⇒ || ||S S for all S S. Now assume z = z 1. Then there is S = S z, z , (1 + ǫ) ∈ ′ 6 ± ′ · (1 + λ) , such that φ is still a (λ, ǫ)-quasi-isometry, and it holds either z S, then z = d (φ(y), φ(x)) λ d (y, x)+ ǫ = λ + ǫ, but • ∈ || ′|| S ≤ S z > λ + ǫ: contradiction, || ′||S or z S, then d (φ(y), φ(x)) = 1, hence d (y, x) λ + λǫ, but • ′ ∈ S S ≤ z > λ + λǫ: contradiction! || ||S Thus, φ(x) = φ(y) (y 1 x) 1, or (after substitution): φ(yx) = φ(y) x 1. · − · ± · ± The sign might still depend on x and y, which we exclude in the next step. Let c := φ(e), and assume there are x, y G with φ(x) = cx = cx 1, but ∈ 6 − φ(xy) = c (xy) 1 = cxy. Then − 6 1 a α c (xy)− = φ(xy) = φ(x) y = cxy · α 1 1 2 for some α = 1, hence xy = y− x− . If α = +1, we have (xy) = e, and ± y 1 hence φ(xy) = c (xy). If α = 1, we have x = x− , which contradicts our 1 − 1 1 premise, unless x = x− . However, if x = x− , we have φ(x) = cx− . We conclude that φ(x) = cx for all x G, or φ(x) = cx 1 for all x G. The ∈ − ∈ latter case leads to

1 1 1 β β cyx = φ x− y− = φ x− y = cxy

x 1 for all x,y G, and someβ = 1. Again, the case β = 1 leads to y = y− , ∈ ± 1 − which we excluded, unless y = y− . So both cases for β lead to the conclusion Shared Rough and Quasi-Isometries 89 that G must be abelian. Indeed, in the abelian case, the inversion is a shared isometry of all symmetric ﬁnite generating systems, and it is non-trivial if and only if G is not of exponent 2.

Hence, UQIsomS(G) ∼= URIsomS(G) ∼= G ⋊ C2 if and only if G is abelian and not of exponent 2, UQIsomS(G) ∼= URIsomS(G) ∼= G otherwise.

Example 146 ...... 146 No ﬁnite group has Property 145, as its diameter is limited. Torsion in itself is an obstruction to it: Let G have Property 145, then each element of G is either torsionsfree, or of exponent 1, 2, 3, 4, or 6 – these are those arguments for which the Euler totient function ϕ is 2 or less ([S]): Let x G be an element n ∈ with x = e. If ϕ(n) > 2, we can choose two diﬀerent generators a, b of Cn, and hence xa and xb are powers of each other, and yet (xa) = (xb) 1. Still, 6 ± there might be other optimal or quasi-optimal families for these groups.

Example 147 ...... 147 Let F be the free group generated by S with #S = n 2. Let g, h F , n 0 0 ≥ ∈ n g = h 1, and R N be arbitrary. Assume h is not a power of g and not 6 ± ∈ ∗ neutral (otherwise switch them; both cannot happen as Fn is torsionfree). If g = e, choose x S such that h is not a power of x, otherwise let x = g. Let ∈ 0 P be the maximum of R and the wordlength of h in S0. Deﬁne

j S(g, h, R) := x x (P +1) s s S r x , j = 1,... #(S r x ) . { } ∪ j | j ∈ 0 { } 0 { } n o The exponents (P + 1)j are chosen such that any non-trivial product of the (P +1)j elements x sj has large enough wordlength in S0, that it cannot equal h, at least for the ﬁrst R steps in the Cayley graph. After this, the powers (P +1)j x successively become available and “free” the generators sj to generate each other element of F , such that h R. The family S of all these n || ||S ≥ generating systems is quasi-optimal due to Lemma 145.

Example 148 ...... 148 In a similar way, we may deﬁne quasi-optimal generating families for free abelian groups. We give the explicit example for G = Z (written additively): Again, assume h is not neutral and not a multiple of g. If g is zero, let P = 1+(R h ), otherwise choose P 1 + (R h g ) and coprime to ∨ | | ≥ ∨ | | ∨ | | g. Then deﬁne S(g, h, R) := g, P 2, P 3 + 1 . { } For arbitrary f.g. free abelian groups, do this componentwise.

The “delayed generation method” we applied in Examples 147 and 148 can sometimes be generalized to other f.g. groups: Choose a ﬁnite generating system S0, then ﬁnd a suitable element x G such that x and g together do not generate P P 2 P 3 ∈ h. Add x s1, x s2, x s3 and so on, after choosing P large enough and taking 90 Claim 150

P care for the group’s relations: If e.g. holds x s1 = s3, choose P even larger, or change the sequence of the generators.

Question Is there a torsion-free group without Property 145, or which does not admit an optimal generating family?

Deﬁnition 149 ...... 149 Let G and H be f.g. groups, and let SH be a family of generating systems of H. We call a pair of maps η : G H and η : H G an S -semi-shared → ′ → H quasi-isometry if there are λ, ǫ 0 with: ≥ For each S S there is a ﬁnite generating system S of G which • H ∈ H G makes (η, η ) : Cay(G, S ) Cay(H, S ) a (λ, ǫ)-quasi-isometry. ′ G → H When we speak of an “S -semi-shared quasi-isometry η : G H” a suitable H → η′ shall always be implied.

Theorem 150 150 y 1 Let G and H be f.g. groups with x = x− for all x, y G (resp. H), unless 1 6 ∈ x = x− . Let G and H be non-abelian, or of exponent 2. Let SG and SH be quasi-optimal families of G and H, respectively, and let η : G H be an S - → H semi-shared quasi-isometry, such that η′ is an SG-semi-shared quasi-isometry. Then G and H are isomorphic as groups.

Proof We first note that η η : H H is an S -uniform quasi-isometry, ◦ ′ → H and hence it is given by multiplication from the left with an element c H. ∈ Consider η : G H given by h c 1 η(h). Then (η , η ) is another S - ′′ → 7→ − · ′′ ′ H semi-shared quasi-isometry, (η′, η′′) is a SG-semi-shared quasi-isometry, and η η is the identity. ′′ ◦ ′ Conversely, η η : G G also is a multiplication from the left with an element ′ ◦ ′′ → c G. We easily find ′ ∈ 2 (η′ η′′ η′ η′′)(h) = (c′) h · = (η′ id η′′)(h) = c′ h · H · · for all h H, thus c = e, and consequently η η is the identity as well. ∈ ′ ′ ◦ ′′ Without loss of generality, and to ease our notation, we may assume that (η, η′) already fulfills η η = id and η η = id . ◦ ′ H ′ ◦ G Now define

ξ : UQIsomS (H) UQIsomS (G) H → G φ η′ φ η, and 7→ ◦ ◦ ξ′ : UQIsomS (G) UQIsomS (H) G → H ψ η ψ η′. 7→ ◦ ◦ Shared Rough and Quasi-Isometries 91

ξ is well-deﬁned: For each S S choose S S such that (η, η ) is a G ∈ G H ∈ H ′ uniform quasi-isometry. Then for each φ UQIsomS (H), the map η φ ∈ H ′ ◦ ◦ η : G G is a uniform quasi-isometry as well; the same accounts for ξ . → ′ Furthermore, we have

ξ(φ ) ξ(φ ) = η′ φ η η′ φ η = ξ(φ φ ) 1 ◦ 2 1 2 1 ◦ 2 ξ′(ψ ) ξ′(ψ ) = η ψ η′ η ψ η′ = ξ′(ψ ψ ) 1 ◦ 2 1 2 1 ◦ 2 ξ′(ξ(φ)) = η η′ φ η η′ = φ

ξ(ξ′(ψ)) = η′ η ψ η′ η = ψ for all g G. This means that (ξ, ξ ) constitutes an isomorphism between ∈ ′ UQIsomS (H) = H and UQIsomS (G) = G. H ∼ G ∼

A similar theorem should hold in the abelian case. We now want to weaken the hypothesis of Theorem 150, with the goal of a suﬃcient criterion for commensurability. For this, we will rework the proof of Lemma 145, which yields a generalized form of homomorphism.

Lemma 151 ...... 151 Let G and H be f.g. groups, and SH a family of generating systems of H satisfying Property 145. Let φ : G H be an S -semi-shared quasi-isometry → H with φ(e) = e. Then φ fulﬁlls φ(gh) = φ(g) φ(h) 1 for all g, h G, where · ± ∈ the sign might depend on g and h.

Proof Let x, y G be arbitrary, z = y 1 x, and z = φ(y) 1 φ(x). Assume ∈ − ′ − z = φ(z) 1. Then we may choose S = S z , φ(z), (λ2 + 1) (ǫ + 1) S ′ 6 ± H ′ · ∈ H a suitable generating system to separate z from φ(z). Then we have ′

z′ = dS (φ(y), φ(x)) SH H

λ dSG (y, x) + ǫ = λ dSG (e, z) + ǫ ≤ · · λ2 d (φ(e), φ(z)) + λ2 ǫ + ǫ ≤ · SH = λ2 φ(z) + λ2 ǫ + ǫ, · SH and, similarly:

2 2 φ(z) λ z′ + λ ǫ + ǫ, SH ≤ · SH

Now one of z and φ ( z) is 1, while the other is larger than λ2 + λ2 ǫ + ǫ, || ′||SH || ||SH contradiction. Hence, z is φ(z)α for some suitable α = 1, which depends on ′ ± x and y. Substituting y = g and z = h yields φ(gh) = φ(g) φ(h) 1. · ±

Note that it is always possible to switch from an arbitrary semi-shared quasi- isometry φ to one with φ(e) = e by a simple translation. The translation even preserves the constants λ and ǫ of the quasi-isometry. 92 Claim 153

Theorem 152 152 Let G, H, and φ : G H be as in Lemma 151. → Assume one of the following statements holds:

1. G admits a generating system S such that φ(s) 2 = e for each s S. ∈ 2. G admits a generating system S such that:

(a) There is no x φ(S S 1), with x2 = e. ∈ ∪ − (b) There are no x, y φ(S S 1), x = y 1, with x2 = y2. ∈ ∪ − 6 ± (c) There are no x, y φ(S S 1), x = y 1, with (xy)2 = e. ∈ ∪ − 6 ± (d) There are no x, y φ(S S 1), x = y 1, with xy = x. ∈ ∪ − 6 ± (e) There are no x, y φ(S S 1), x = y 1, with xy = x 1. ∈ ∪ − 6 ± − (f) There are at least two distinct elements in S, which are not inverses of each other.

(In particular, G is not abelian.)

Then G and H are commensurable up to ﬁnite kernels (Deﬁnition 123).

Proof Due to Lemma 151 we have in each case

φ(g h) = φ(g) φ(h)σ(g,h) · with σ(g, h) 1 for any g G and h H. Observe that σ(g, e) = σ(e, g) ∈ {± } ∈ ∈ = +1. If φ(h) is neutral or of order 2, we choose σ(g, h) to be +1 without loss of generality. We next show that under both hypothesis φ must be a homomorphism. Due to Proposition 125 G and H then must be commensurable up to ﬁnite kernels. (1) We trivially have

φ(gs) = φ(g) φ(s) · for any g G and s S. By induction, φ must be a homomorphism. ∈ ∈ (2) Let g G and s, t S be arbitrary, s = t 1. We make use of the associa- ∈ ∈ 6 ± tive law:

φ(gst) = φ(g) φ(s)α φ(t)β · · = φ(g) φ(s) φ(t)γ δ · · for some α, β, γ, δ = 1. The sixteen possible cases resolve as in Table 3.1. ± Fourteen cases subsequently contradict our premise. Both remaining cases 1 and 7 demand α = σ(g, s) = +1, for all g G and s S, so we have ∈ ∈ φ(gs) = φ(g) φ(s), · and, again by induction, φ must be a homomorphism. Shared Rough and Quasi-Isometries 93

Nr α β γ δ xα yβ = (x yγ)δ contradiction · · 1 + + + + – no 2 + + + (xy)2 = e yes (c) − 3 + + + y2 = e yes (a) − 4 + + xy = x 1 yes (e) − − − 5 + + + y2 = e yes (a) − 6 + + xy = x 1 yes (e) − − − 7 + + – no − − 8 + (xy 1)2 = e yes (c) − − − − 9 + + + x2 = e yes (a) − 10 + + yx = y 1 yes (e) − − − 11 + + x2 = y2 yes (b) − − 12 + xy = x yes (d) − − − 13 + + x 2 = y2 yes (b) − − − 14 + xy = x yes (d) − − − 15 + x2 = e yes (a) − − − 16 yx = y 1 yes (e) − − − − − Table 3.1: The sixteen cases of the proof of Theorem 152.3. For convenience, we use x = φ(s) and y = φ(t).

Corollary 153 ...... 153 Let G, H, and φ be as in Theorem 152, and let H be non-abelian, or of exponent 2. In addition, xy = x 1 shall hold for all x, y H with x = x 1. Then H 6 − ∈ 6 − is the quotient of G by the ﬁnite subgroup ker φ E G.

Proof Lemma 145 ensures that φ φ : H H is given by multiplication ◦ ′ → with a ﬁxed element of H, and in particular, φ must be surjective. From the proof of Theorem 152 we know that φ is a homomorphism with ﬁnite kernel. Using the First Isomorphism Theorem ([Bo], Korollar 1.2.7), we see H = im φ ∼= G/ ker φ.

Unfortunately, Proposition 134 is not yet strong enough to constitute a reversal of Corollary 153. Still, we are confident to soon find a sustainable connection between semi-shared quasi-isometries and quotients of finite kernel. Through the means of residual finiteness, it might then be possible to finally find a perfectly fitting geometrical equivalence relation which equals commensurability—which was our motivation for this chapter. 94 Bibliography

[AP] N. Aronszajn, P. Panitchpakdi, Extension of Uniformly Continuous Transformations and Hyperconvex Metric Spaces, Paciﬁc J. Math. 6, (1956) 405–439, http://projecteuclid.org/handle/euclid.pjm/1103043960

[Ba] R. Baer, Endlichkeitskriterien f¨ur Kommutatorgruppen, Math. Annalen 124 (1952) 161–177

[BBI] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Grad- uate Studies in Mathematics Volume 33, AMS

[Bi1] G. Birkhoﬀ, Lattice Theory, American Mathematical Society Colloquium Publications Vol. XXV, 2nd ed. (1948) and 3rd ed. (1960)

[Bi2] G. Birkhoﬀ, Von Neumann and Lattice Theory, Bull. Amer. Math. Soc. 64, Nr 3, Part 2 (1958) 50–56, http://www.ams.org/bull/1958-64-03/S0002-9904-1958-10192-5/ S0002-9904-1958-10192-5.pdf

[Bo] S. Bosch, Algebra, 4th edition, Springer-Verlag (2001)

[dH] P. de la Harpe, Topics in Geometric Group Theory, The University of Chicago Press (2000)

[EK] R. Esp´ınola, M. A. Khamsi, Introduction to Hyperconvex Spaces, ap- peared in Handbook of Metric Fixed Point Theory by W. A. Kirk and B. Sims, Kluwer 2001, http://drkhamsi.com/publication/Es-Kh.pdf

[FM] B. Farb, L. Mosher, On the asymptotic geometry of abelian-by-cyclic groups, Acta Mathematica 184, 2 (2000) 145–202, preprint available at http://arxiv.org/abs/math.GR/0005181

[GKM] T. Gelander, A. Karlsson, G. A. Margulis, Superrigidity, Generalized Harmonic Maps and Uniformly Convex Spaces, Geom. funct. anal. 17 (2008) 1524–1550, preprint at http://arxiv.org/abs/math/0606256

[Gl] V. Glivenko, Géometrie des systèmes de chosen normées, Am. Jour. of Math. 58 (1936) 799–828

[Gr] G. A. Gr¨atzer, General Lattice Theory, 2nd ed., Birkh¨auser Verlag (2003) 96

[Gv1] M. Gromov, Hyperbolic manifolds, groups and actions, Ann. of Math. Studies 97, Princeton Univ.Press, Princeton (1981) 183–213

[Gv2] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkh¨auser (1999)

[Gv3] M. Gromov, Asymptotic Invariants of Inﬁnite Groups in Geometric Group Theory, Volume 2, LMS Lecture Notes Series 182, Cambridge University Press

[He] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer NY (2001)

[Ho] H. S. Holdgr¨un, Analysis, Band 2, Leins Verlag G¨ottingen (2001)

[HR] D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992) 125–153, http://www.digizeitschriften.de/index.php?id=166&ID=379571

[HU] D. H. Hyers, S. M. Ulam, Approximate Isometries of the Space of Con- tinuous Functions, Ann. of Math. 48 (2) (1947) 285–289

[Hy] D. H. Hyers, On the Stability of the Linear Functional Equation, Proc. Nat. Acad. Sci. USA, 27 (1941) 222–224, http://www.pnas.org/content/27/4/222.full.pdf+html

[I] J. R. Isbell, Six theorems about injective metric spaces, Commentarii mathematici Helevetici 39 (1964) 65–76, http://www.digizeitschriften.de/resolveppn/GDZPPN002058340

[Kn] M. Kanai, Rough Isometries, and combinatorial approximations of geometries of non-compact riemannian manifolds, J. Math. Soc. Japan 37 (1985) 391–413

[Kp] I. Kaplansky, Lattices of continuous functions, Bull. Am. Math. Soc. 53 (1947) 617–622

[L] C. H. Li, On isomorphisms of ﬁnite Cayley graphs — a survey, Discrete Mathematics 256 (2002) 301–334

[Me] K. Menger, Untersuchungen ¨uber allgemeine Metrik, Mathematische Annalen 100 (1928) 75–163, http://www.digizeitschriften.de/resolveppn/PPN235181684_0100

[Mn] B. Monjardet, Metrics on partially ordered sets — a survey, Discrete Mathematics 35 (1981) 173–184

[Mo] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies 78, Princeton Univ.Press, Princeton (1973) 183–213

[MU] S. Mazur, S. Ulam, Sur les transformations isom´etriques d’espaces vec- toriels norm´es, C. R. Acad. Sci., Paris 194 (1932), 946–948 BIBLIOGRAPHY 97

[P] P. Petersen V, Gromov-Hausdorﬀ Convergence of Metric Spaces, AMS Proc. Symp. in Pure Math Math. 54 (3) (1993) 489–504

[Ra1] T. M. Rassias, Isometries and approximate isometries, Internat. J. Math. and Math. Sciences, 25 (2) (2001) 73–91, http://www.hindawi.com/GetArticle.aspx?doi=10.1155/ S0161171201004392

[Ra2] T. M. Rassias, A New Generalization of a Theorem of Jung for the Orthogonality Equation, Applicable Analysis, 81 (2002) 163–177, and: V. A. Fa˘ıziev, T. M. Rassias, P. K. Sahoo, The Space of (ψ, γ)-additive Mappings on Semigroups, Trans. AMS, 354 Number 11 (2002) 4455– 4472, http://www.ams.org/tran/2002-354-11/ S0002-9947-02-03036-2/S0002-9947-02-03036-2.pdf

[Re] M. Requardt, Continuum Limit of Discrete Geometries, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 285

[Ro] R. T. Rockafellar, Convex Analysis, Princeton University Press (1970)

[S] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, sequence A000010, http://www.research.att.com/~njas/sequences/A000010 [T] A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, Proc. London Math. Soc. Ser. 2, 42 (1937) 230– 265

[U] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York (1960)

[V] J. Väisälä, A proof of the Mazur-Ulam theorem, Amer. Math. Mon. 110, 7 (2003) 633–635, http://www.helsinki.fi/~jvaisala/mazurulam.pdf [vN] J. von Neumann, Lectures on continuous geometries, Princeton 1936- 1937 (2 vols.), in particular chapter XVII

[Wr] D. Werner, Funktionalanalysis, Springer (2005)

[Wv] N. Weaver, Lipschitz Algebras, World Scientiﬁc (1999)